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MATHEMATICS
VIBRANT ACADEMY (India) Private Limited
B-41, Road No.2, Indraprastha Industrial Area, Kota-324005 (Raj.)Tel. : 06377791915, (0744) 2778899, Fax : (0744) 2423405
Email: admin@vibrantacademy.com Website : www.vibrantacademy.comWebsite : dlp.vibrantacademy.com
Believe In Excellence
CONTENTSKEY CONCEPTS — Page-2-7
PROFICIENCY TEST — Page-8-10
EXERCISE-I — Page-11-12
EXERCISE-II — Page-12-13
EXERCISE-III — Page-14-15
EXERCISE-IV — Page-15-18
EXERCISE-V — Page-18-24
ANSWER KEY — Page-25-26
MATRICES
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KEY CONCEPTSMATRICES
USEFUL IN STUDY OF SCIENCE, ECONOMICS AND ENGINEERING
1. Definition : Rectangular array of m n numbers . Unlike determinants it has no value.
A =
a a a
a a a
a a a
n
n
m m mn
11 12 1
21 22 2
1 2
......
......
: : : :
......
or
a a a
a a a
a a a
n
n
m m mn
11 12 1
21 22 2
1 2
......
......
: : : :
......
Abbreviated as : A = a i j 1 i m ; 1 j n, i denotes the row and
j denotes the column is called a matrix of order m × n.
2. Special Type Of Matrices :(a) Row Matrix : A = [ a
11 , a
12 , ...... a
1n ] having one row . (1 × n) matrix.
(or row vectors)
(b) Column Matrix : A =
a
a
a m
11
21
1
:
having one column. (m × 1) matrix(or column vectors)
(c) Zero or Null Matrix : (A = Om n
)
An m n matrix all whose entries are zero .
A =
0 0
0 0
0 0
is a 3 2 null matrix & B =
0 0 0
0 0 0
0 0 0
is 3 3 null matrix
(d) Horizontal Matrix : A matrix of order m × n is a horizontal matrix if n > m.
1152
4321
(e) Verical Matrix : A matrix of order m × n is a vertical matrix if m > n.
42
63
11
52
(f) Square Matrix : (Order n)
If number of row = number of column a square matrix.
Note (i) In a square matrix the pair of elements aij & a
j i are called Conjugate Elements .
e.g.a a
a a11 12
21 22
(ii) The elements a11
, a22
, a33
, ...... ann
are called Diagonal Elements . The line along whichthe diagonal elements lie is called " Principal or Leading " diagonal.
The qty ai i
= trace of the matrice written as , i.e. tr A
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Square Matrix
Triangular Matrix Diagonal Matrix denote asd
dia (d
1 , d
2 , ....., d
n) all elements
except the leading diagonal are zero
diagonal Matrix Unit or Identity Matrix
Note: Min. number of zeros in a diagonal matrix of order n = n(n – 1)
"It is to be noted that with square matrix there is a corresponding determinant formed by the elements of A in thesame order."
3. Equality Of Matrices :Let A = [a
i j ] & B = [b
i j ] are equal if ,
(i) both have the same order . (ii) ai j
= b i j for each pair of i & j.
4. Algebra Of Matrices :
Addition : A + B = a bi j i j where A & B are of the same type. (same order)
(a) Addition of matrices is commutative.
i.e. A + B = B + A A = m n ; B = m n(b) Matrix addition is associative .
(A + B) + C = A + (B + C) Note : A , B & C are of the same type.(c) Additive inverse.
If A + B = O = B + A A = m n5. Multiplication Of A Matrix By A Scalar :
If A =
bacacbcba
; k A =
bkakckakckbkckbkak
6. Multiplication Of Matrices : (Row by Column)AB exists if , A = m n & B = n p
2 3 3 3
AB exists , but BA does not AB BA
Note : In the product AB , A prefactor
B post factor
A = (a1 , a
2 , ...... a
n) & B =
n
2
1
b:
bb
1 n n 1A B = [a
1 b
1 + a
2 b
2 + ...... + a
n b
n]
If A = a i j m n & B = bi j n p matrix , then (A B)i j
= r
n
1 a
i r . b
r j
A = 1 3 2
0 2 4
0 0 5
; B =
1 0 0
2 3 0
4 3 3
Upper Triangular Lower Triangular a
i j = 0 i > j a
i j = 0 i < j
Note that : Minimum number of zeros ina triangular matrix oforder n = n(n–1)/2
d
d
d
1
2
3
0 0
0 0
0 0
a i j
= 1
0
if i j
if i j
If d1 = d
2 = d
3 = a Scalar Matrix
If d1 = d
2 = d
3 = 1 Unit Matrix
Note: (1) (2)
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SProperties Of Matrix Multiplication :1. Matrix multiplication is not commutative .
A =
0011
;B =
0001
; AB =
0001
; BA =
0011
AB BA (in general)
2. AB =
2211
1111
=
0000
AB = O A = O or B = O
Note: If A and B are two non- zero matrices such that AB = O then A and B are called the divisors of zero.Also if [AB] = O | AB | | A | | B | = 0 | A | = 0 or | B | = 0 but not the converse.If A and B are two matrices such that(i) AB = BA A and B commute each other
(ii) AB = – BA A and B anti commute each other3. Matrix Multiplication Is Associative :
If A , B & C are conformable for the product AB & BC, then(A . B) . C = A . (B . C)
4. Distributivity :
A B C AB AC
A B C AC BC
( )
( )
Provided A, B & C are conformable for respective productss
5. POSITIVE INTEGRAL POWERS OF A SQUARE MATRIX :For a square matrix A , A2 A = (A A) A = A (A A) = A3 .Note that for a unit matrix I of any order , Im = I for all m N.
6. MATRIX POLYNOMIAL :If f (x) = a
0xn + a
1xn – 1 + a
2xn – 2 + ......... + a
nx0 then we define a matrix polynomial
f (A) = a0An + a
1An–1 + a
2An–2 + ..... + a
nIn
where A is the given square matrix. If f (A) is the null matrix then A is called the zero or root of the polynomialf (x).
DEFINITIONS :(a) Idempotent Matrix : A square matrix is idempotent provided A2 = A.
Note that An = A n > 2 , n N.
(b) Nilpotent Matrix: A square matrix is said to be nilpotent matrix of order m, m N, if
Am = O , Am–1 O.
(c) Periodic Matrix : A square matrix is which satisfies the relation AK+1 = A, for some positive integer K, is aperiodic matrix. The period of the matrix is the least value of K for which this holds true.Note that period of an idempotent matrix is 1.
(d) Involutary Matrix : If A2 = I , the matrix is said to be an involutary matrix.Note that A = A–1 for an involutary matrix.
7. The Transpose Of A Matrix : (Changing rows & columns)Let A be any matrix . Then , A = a
i j of order m n
AT or A = [ aj i ] for 1 i n & 1 j m of order n m
Properties of Transpose : If AT & BT denote the transpose of A and B ,(a) (A ± B)T = AT ± BT ; note that A & B have the same order.
IMP. (b) (A B)T = BT AT A & B are conformable for matrix product AB.(c) (AT)T = A(d) (k A)T = k AT k is a scalar .
General : (A1 , A
2 , ...... A
n)T = An
T , ....... , AT2 , AT
1 (reversal law for transpose)
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S8. Symmetric & Skew Symmetric Matrix :
A square matrix A = a i j is said to be ,
symmetric if ,a
i j= a
j i i & j (conjugate elements are equal) (Note A = AT)
Note: Max. number of distinct entries in a symmetric matrix of order n is 2
)1n(n .
and skew symmetric if , a
i j= a
j i i & j (the pair of conjugate elements are additive inverse
of each other) (Note A = –AT )Hence If A is skew symmetric, then
ai i
= ai i a
i i= 0 i
Thus the digaonal elements of a skew symmetric matrix are all zero , but not the converse .
Properties Of Symmetric & Skew Matrix :P 1 A is symmetric if AT = A
A is skew symmetric if AT = A
P 2 A + AT is a symmetric matrix
A AT is a skew symmetric matrix .Consider (A + AT)T = AT + (AT)T = AT + A = A + AT
A + AT is symmetric .Similarly we can prove that A AT is skew symmetric .
P 3 The sum of two symmetric matrix is a symmetric matrix andthe sum of two skew symmetric matrix is a skew symmetric matrix .Let AT = A ; BT = B where A & B have the same order .
(A + B)T = A + BSimilarly we can prove the other
P 4 If A & B are symmetric matrices then ,(a) A B + B A is a symmetric matrix(b) AB BA is a skew symmetric matrix .
P 5 Every square matrix can be uniquely expressed as a sum of a symmetric and a skew symmetric matrix.
A = 1
2 (A + AT) +
1
2 (A AT)
P QSymmetric Skew Symmetric
9. Adjoint Of A Square Matrix :
Let A = a i j =
333231
232221
131211
aaaaaaaaa
be a square matrix and let the matrix formed by the cofactors
of [ai j ] in determinant A is =
333231
232221
131211
CCCCCCCCC
.
Then (adj A) =
332313
322212
312111
CCCCCCCCC
V. Imp. Theorem : A (adj. A) = (adj. A).A = |A| In , If A be a square matrix of order n.
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SNote : If A and B are non singular square matrices of same order, then(i) | adj A | = | A |n – 1
(ii) adj (AB) = (adj B) (adj A)(iii) adj(KA) = Kn–1 (adj A), K is a scalar
Inverse Of A Matrix (Reciprocal Matrix) :A square matrix A said to be invertible (non singular) if there exists a matrix B such that,
A B = I = B A
B is called the inverse (reciprocal) of A and is denoted by A 1 . Thus
A 1 = B A B = I = B A .
We have , A . (adj A) = A In
A 1 A (adj A) = A 1 In
In
(adj A) = A 1 A In
A 1 = ( )
| |adj A
A
Note : The necessary and sufficient condition for a square matrix A to be invertible is that A 0.
Imp. Theorem : If A & B are invertible matrices ofthe same order , then (AB) 1 = B 1 A 1. This is reversal law forinverse.
Note :(i) If A be an invertible matrix , then AT is also invertible & (AT) 1 = (A 1)T.
(ii) If A is invertible, (a) (A 1) 1 = A ; (b) (Ak) 1 = (A 1)k = A–k, k N
(iii) If A is an Orthogonal Matrix. AAT = I = ATA
(iv) A square matrix is said to be orthogonal if , A 1 = AT .
(v) | A–1 | = |A|
1
SYSTEM OF EQUATION & CRITERIAN FOR CONSISTENCYGAUSS - JORDAN METHOD
x + y + z = 6x y + z = 2
2 x + y z = 1
or
zyx2zyxzyx
=
126
112111111
zyx
=
126
A X = B A 1 A X = A 1 B
X = A 1 B = |A|
B).A.adj(.
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SNote :(1) If A 0, system is consistent having unique solution
(2) If A 0 & (adj A) . B O (Null matrix) ,
system is consistent having unique non trivial solution .
(3) If A 0 & (adj A) . B = O (Null matrix) ,system is consistent having trivial solution .
(4) If A= 0 , matrix method fails
If (adj A) . B = null matrix = O If (adj A) . B O
Consistent (Infinite solutions) Inconsistent (no solution)
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PROFICIENCY TEST-011. In the following, upper triangular matrix is
(A)
303
020
001
(B)
100
300
245
(C)
400
320(D)
00
30
12
2. If
01
25A and
1–5
32B , then |2A – 3B| equals
(A) 77 (B) –53 (C) 53 (D) –77
3. For a square matrix A = [aij], aij = 0, when i j, then A is
(A) unit matrix (B) scalar matrix (C) diagonal matrix (D) None of these
4. If A and B are matrices of order m × n and n × n respectively, then which of the following are defined
(A) AB, BA (B) AB, A2 (C) A2, B2 (D) AB, B2
5. If
213
201–A and
103
72
51–
B , then
(A) AB and BA both exist (B) AB exists but not BA
(C) BA exists but not AB (D) Both AB and BA do not exist
6. If A is a matrix of order 3 × 4, then both ABT and BTA are defined if order of B is
(A) 3 × 3 (B) 4 × 4 (C) 4 × 3 (D) 3 × 4
7. Matrix
011–7
1105–
7–50
is a
(A) Diagonal matrix (B) Upper triangular matrix
(C) Skew-symmetric matrix (D) Symmetric matrix
8. If A is symmetric as well as skew symmetric matrix, then
(A) A is a diagonal matrix (B) A is a null matrix
(C) A is a unit matrix (D) A is a triangular matrix
9. If A is symmetric matrix and B is a skew-symmetric matrix, then for n N, false statement is
(A) An is symmetric when n is odd (B) An is symmetric only when n is even
(C) Bn is skew symmetric when n is odd (D) Bn is symmetric when n is even
10. Let A be a square matrix. Then which of the following is not a symmetric matrix
(A) A + AT (B) AAT (C) ATA (D) A – AT
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11. If
11
11A and n N, then AAn is equal to
(A) 2nA (B) 2n – 1 A (B) nA (D) None of these
12. If A = [aij] is scalar matrix of order n × n such that aii = k for all i, then |A| equals
(A) nk (B) n + k (C) nk (D) kn
13. If
1–1
4–3A , then for every positive integer n, AAn is equal to
(A) 1 2n 4n 8
n 1 2n
(B)
n2–1n
n4–n21(C)
1– 2n 4n
n n 2
(D) None of these
14. If A is any skew-symmetric matrix of odd orders, then |A| equals(A) –1 (B) 0 (C) 1 (D) None of these
15. If
0a–b
a0c–
b–c0
A and
2
2
2
cbcac
bcbab
acaba
B then AB is equal to
(A) A (B) B (C) an Identity matrix (D) a Null matrix
PROFICIENCY TEST-02
1. The root of the equation [x 1 2] 0
1
1–
x
011
101
110
is
(A) 3
1(B)
3
1– (C) 0 (D) 1
2. For square matrices A and B, AB = O, then {O : null matrix}
(A) A = O or B = O (B) A = O and B = O
(C) It is not necessary that A = O and/or B = O (D) None of these
3. If A and B are matrices of order m × n and n × m respectively, then the order of matrix BT (AT)T is
(A) m × n (B) m × m (C) n × n (D) Not defined
4. If
200
432
321
A , then the value of adj (adj A) is
(A) 4A2 (B) –2A (C) 2A (D) A2
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5. If
xcosxsin–
xsinxcosA and A.(adj A) = k
10
01 , then k equals
(A) sinx cosx (B) 1 (C) sin 2x (D) –1
6. If
513–
1–04
32–1
A , then (adj A)23 =
{i.e., the element of (adj A) which belongs to second row and third column}(A) 13 (B) –13 (C) 5 (D) –5
7. (adj AT) – (adj A)T equals(A) |A| I (B) 2|A| I (C) Null matrix (D) Unit matrix
8. If
10
01C,
32
64B,
31
32A , then which of these matrices are invertible ?
(A) A and B (B) B and C (C) A and C (D) All
9. Which of the following matrices is inverse of itself
(A)
111
111
111
(B) 3 2
4 3
(C)
101
000
101
(D)
010
111
010
10. If D is a diagonal matrix with diagonal elements as {d1, d2, d3 ..., dn} in order, then we may represent it asD = diag (d1, d2, ......., dn). Then Dn equals(A) D (B) diag (d1
n – 1, d2n – 1, ......, dn
n – 1)(C) diag (d1
n, d2n, ......, dn
n) (D) None of these
11. If
100
0cossin
0sin–cos
A , then
(A) adj A = A (B) adj A = A–1 (C) A–1 = –A (D) None of these
12. If A is invertible matrix, then det (A–1) equals {where, det (B) means determinant of matrix B}
(A) det (A) (B) 1
det (A) (C) 1 (D) None of these
13. If A and B are non-zero square matrices of the same order such that AB = O, then {O : null matrix}(A) Either adj A = O or adj B = O (B) adj A = O and adj B = O(C) Either |A| = 0 or |B| = 0 (D) |A| = 0 and |B| = 0
14. Let A be an idempotent square matrix, then (I + A)4 is :(A) I – A (B) I + A (C) I + 15A (D) I
15. If A and B are two square matrices such that B = –A–1BA, then (A + B)2 =(A) A2 + 2BA + B2 (B) A2 + B2 (C) A2 + 2AB + B2 (D) A2 – B2
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EXERCISE-I1. Find the number of 2 × 2 matrix satisfying
(i) aij is 1 or –1 ; (ii) 2
11a + 212a = 2
21a + 222a = 2 ; (iii) a
111 a
21 + a
12 a
22 = 0
2. Find the value of x and y that satisfy the equations.
420323
xxyy
=
1010y3y333
3. Let A =
dcba
and B =
qp
00
. Such that AB = B and a + d = 5050. Find the value
of (ad – bc).
4. Define A =
0310
. Find a vertical vector V such that (A8 + AA6 + A4 + A2 + I)V =
110
(where I is the 2 × 2 identity matrix).
5. If, A =
302120201
, then show that the maxtrix A is a root of the polynomial f (x) = x3 – 6x2 + 7x + 2.
6. If the matrices A =
4321
and B =
dcba
(a, b, c, d not all simultaneously zero) commute, find the value of bca
bd
. Also show that the
matrix which commutes with A is of the form
32
7. If
a1cba
is an idempotent matrix. Find the value of f(a), where f(x) = x– x2, when bc = 1/4. Hence
otherwise evaluate a.
8. If the matrix A is involutary, show that 2
1(I + A) and
2
1(I – A) are idempotent and
2
1(I + A)·
2
1(I – A)=O.
9. Show that the matrix A =
1201
can be decomposed as a sum of a unit and a nilpotent marix. Hence
evaluate the matrix 2007
1201
.
10. Given matrices A =
3y1y2x1x1
; B =
13z323
z33
Obtain x, y and z if the matrix AB is symmetric.
11. Let X be the solution set of the equation Ax = I, where A =
433434110
and I is the corresponding unit
matrix and x N then find the minimum value of )sin(cos xx , R.
12. A =
28bc521a3
is Symmetric and B =
f62cb2eab
a3d is Skew Symmetric, then find AB.
Is AB a symmetric, Skew Symmetric or neither of them. Justify your answer.
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13. A is a square matrix of order n.l = maximum number of distinct entries if A is a triangular matrixm = maximum number of distinct entries if A is a diagonal matrixp = minimum number of zeroes if A is a triangular matrixIf l + 5 = p + 2m, find the order of the matrix.
14. If A is an idempotent non zero matrix and I is an identity matrix of the same order, find the value of n, n
N, such that ( A + I )n = I + 127 A.
15. Consider the two matrices A and B where A =
3421
; B =
35
. If n(A) denotes the number of elementss
in A such that n(XY) = 0, when the two matrices X and Y are not conformable for multiplication. If C =
(AB)(B'A); D = (B'A)(AB) then, find the value of
)B(n)A(n
)D(n|D|)C(n 2
.
EXERCISE-II
1. A3 × 3
is a matrix such that | A | = a, B = (adj A) such that | B | = b. Find the value of (ab2 + a2b + 1)S
where S2
1 = ......
b
a
b
a
b
a5
3
3
2
up to , and a = 3.
2. For the matrix A =
433332
544 find AA–2.
3. Given A =
132142111
, B =
4332
. Find P such that BPA =
010101
4. Given the matrix A =
531531
531 and X be the solution set of the equation AAx = A,
where x N – {1}. Evaluate
1x
1x3
3
where the continued product extends x X.
5. If F(x) =
1000xcosxsin0xsinxcos
then show that F(x). F(y) = F(x + y)
Hence prove that [ F(x) ]–1 = F(– x).
6. Use matrix to solve the following system of equations.
(i)
6z9y4x4z3y2x
3zyx
(ii)
7z4y3x24z3y2x
3zyx
(iii)
9z4y3x24z3y2x
3zyx
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S7. Let A be a 3 × 3 matrix such that a11 = a33 = 2 and all the other aij = 1. Let A–1 = xA2 + yA + zI then find the
value of (x + y + z) where I is a unit matrix of order 3.
8. Find the matrix A satisfying the matrix equation,
2312
. A .
3523
=
1342
.
9. If A =
n
mk
l and mkn l ; then show that AA2 – (k + n)A + (kn – lm) I = O.
Hence find A–1.
10. Given A=
1212
; B=
1339
. I is a unit matrix of order 2. Find all possible matrix X in the following cases.
(i) AX = A (ii) XA = I (iii) XB = O but BX O.
11. If A =
4221
then, find a non-zero square matrix X of order 2 such that AX = O. Is XA = O.
If A =
3221
, is it possible to find a square matrix X such that AX = O. Give reasons for it.
12. Determine the values of a and b for which the system
13b
zyx
a12985123
(i) has a unique solution ; (ii) has no solution and (iii) has infinitely many solutions
13. If A =
4321
; B =
0113
; C =
4221
and X =
43
21xxxx
then solve the following matrix equation.
(a) AX = B – I (b) (B – I)X = IC (c) CX = A
14. If A is an orthogonal matrix and B = AP where P is a non singular matrix then show that the matrix
PB–1 is also orthogonal.
15. Consider the matrices A =
1143
and B =
10ba
and let P be any orthogonal matrix and Q = PAPAPT and
R = PTQKP also S = PBPT and T = PTSKP
Column I Column II
(A) If we vary K from 1 to n then the first row (P) G.P. with common ratio a
first column elements at R will form
(B) If we vary K from 1 to n then the 2nd row 2nd (Q) A.P. with common difference 2
column elements at R will form
(C) If we vary K from 1 to n then the first row first (R) G.P. with common ratio b
column elements of T will form
(D) If we vary K from 3 to n then the first row 2nd column (S) A.P. with common difference – 2.
elements of T will represent the sum of
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EXERCISE-III
1. If
– is square root of 2 (2 × 2 Identity matrix), then , and will satisfy the relation
(A) 1 + 2 + = 0 (B) 1 – 2 + = 0 (C) 1 + 2 – = 0 (D) –1 + 2 + = 0
2. If
cossin–
sincosA , then which of following statement is true
(A) A . A = A and
nn
nnn
cossin–
sincos)A( (B) A . A = A and
ncosnsin–
nsinncos)A( n
(C) A . A = A and
nn
nnn
cossin–
sincos)A( (D) AA . A = A and
ncosnsin–
nsinncos)A( n
3. If
32
21M and M2 – M – I = 0, then equals
(A) –2 (B) 2 (C) –4 (D) 4
4. If A be a matrix such that inverse of 7A is the matrix
7–4
21–, then A equals
(A)
14
21(B)
7/17/2
7/41(C)
12
41(D)
7/17/4
7/21
5. If
01–
10A and (aI + bA)2 = A, (a > 0), then
(A) 2ba (B) 2
1ba (C) 3ba (D)
3
1ba
6. If A and B are square matrices such that AB = B and BA = A, then A2 + B2 is equal to(A) 2AB (B) 2BA (C) A + B (D) None of these
7. If
ab
b–a
1tan–
tan1
1tan
tan–11–
, then
(A) a = sin 2, b = – cos 2 (B) a = cos 2, b = sin 2(C) a = sin 2, b = cos 2 (D) a = cos 2, b = – sin 2
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8. Let the matrices A and B be defined as 1 2
A2 3
and 3 7
B1 3
, then the value of determinant of matrix
(2A7B–1), is :(A) 2 (B) 1 (C) –1 (D) –2
9. There are two possible values of A in the solution of the matrix equation
12A 1 5 A 5 B 14 D
4 A 2A 2 C E F
, where A, B, C, D, E, F are real numbers. The absolute value of the
difference of these two solutions, is :
(A) 13
3(B)
11
3(C)
17
3(D)
19
3
10. If A is a square matrix, and B is a singular matrix of same order, then for a positive integer n, (A–1BA)n equals(A) A–nBnAn (B) AnBnA–n (C) A–1BnA (D) n(A–1BA)
EXERCISE-IV
1. If a b
Ab a
and 2A
, then [AIEEE 2003]
(A) = a2 + b2, = ab (B) = a2 + b2, = 2ab(C) = a2 + b2, = a2 – b2 (D) = 2ab, = a2 + b2
2. Let A =
0 0 1
0 1 0
1 0 0
. The only correct statement about the matrix A is : [AIEEE 2004]
(A) A is a zero matrix (B) A2 = I(C) A–1 does not exist (D) A = –I, where I is a unit matrix
3. Let
1 1 1
A 2 1 3
1 1 1
and
4 2 2
10B 5 0
1 2 3
. If B is the inverse of A, then is : [AIEEE 2004]
(A) –2 (B) 5 (C) 2 (D) –1
4. If A2 – A + I = O, then the inverse of A is : [AIEEE 2005](A) A + I (B) A (C) A – I (D) I – A
5. If 1 0
A1 1
and 1 0
0 1
I , then which one of the following holds for all n 1, by the principle of mathematical
induction [AIEEE 2005](A) An = nA – (n–1)I (B) An = 2n–1A – (n–1)I (C) An = nA + (n –1)I (D) An = 2n–1A + (n–1)I
6. If A and B are square matrices of size n × n such that A2 – B2 = (A – B)(A + B), then which of the following willbe always true ? [AIEEE 2006](A) A = B (B) AB = BA(C) Either A or B is a zero matrix (D) Either A or B is an identity matrix
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7. Let 1 2
A3 4
and a 0
B0 b
, a, b N. Then [AIEEE 2006]
(A) there cannot exists any B such that AB = BA.
(B) there exists more than one but finite numbe of B's such that AB = BA.
(C) there exists exactly one B such that AB = BA.
(D) there exists infinitely many B's such that AB = BA.
8. Let
5 5
A 0 5
0 0 5
. If |A2| = 25, then || equals : [AIEEE 2007]
(A) 52 (B) 1 (C) 1/5 (D) 5
9. Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr(A), the sum of
diagonal entires of A. Assume that A2 = I. [AIEEE 2008]
Statement 1 : If A I and A –I, then detA = –1.
Statement 2 : If A I and A –I, then tr(A) 0.
(A) Statement 1 is false, statement 2 is true.
(B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(D) Statement 1 is true, statement 2 is false.
10. Let A be a 2 × 2 matrix. [AIEEE 2009]
Statement 1 : adj.(adj A) = A
Statement 2 : |adj A| = |A|
(A) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(B) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is false.
(D) Statement 1 is false, statement 2 is true.
11. The number of 3 × 3 non-singular matrices with four entries as 1 and all other entries as 0 is : [AIEEE 2010]
(A) at least 7 (B) less than 4 (C) 5 (D) 6
12. Let A be a 2 × 2 matrix with non-zero entries and let A2 = I, where I is a 2 × 2 identity matrix. Define Tr(A) =
sum of diagonal elements of A and |A| = determinant of matrix A. [AIEEE 2010]
Statement 1 : Tr(A) = 0
Statement 2 : |A| = 1
(A) Statement 1 is false, statement 2 is true.
(B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(D) Statement 1 is true, statement 2 is false.
13. Let A and B two symmetric matrices of order 3. [AIEEE 2011]
Statement 1 : A(BA) and (AB)A are symmetric matrices
Statement 2 : AB is symmetric matrix if matrix multiplication of A with B is commutative.
(A) Statement 1 is false, statement 2 is true.
(B) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.
(C) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.
(D) Statement 1 is true, statement 2 is false.
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14. Let
1 0 0
A 2 1 0
3 2 1
. If u1 and u
2 are column matrices such that 1
1
Au 0
0
and 2
0
Au 1
0
, then u1 + u
2 is
equal to : [AIEEE 2012]
(A)
1
1
0
(B)
1
1
1
(C)
1
1
0
(D)
1
1
1
15. Let P and Q be 3 × 3 matrices P Q. If P3 = Q3 and P2Q = Q2P, then determinant of (P2 + Q2) is equal to :
[AIEEE 2012]
(A) –2 (B) 1 (C) 0 (D) –1
16. If P =
442
331
31
is the adjoint of a 3 × 3 matrix A and |A| = 4, then is equal to :
(A) 0 (B) 4 (C) 11 (D) 5 [JEE Main - 2013]
17. If A is an 3 × 3 non-singular matrix such that AA' = A'A and B = A–1 A', then BB' equals
[JEE Main - 2014]
(A) (B–1)' (B) I + B (C) I (D) B–1
18. If A =
1 2 2
2 1 2
a 2 b
is a matrix satisfying the equation AAT = 9I, where I is 3 × 3 identity matrix, then the
ordered pair (a, b) is equal to: [JEE Main - 2015]
(A) (–2, –1) (B) (2, –1) (C) (–2, 1) (D) (2, 1)
19. If A =
23
ba5 and A adj A = A AAT, then 5a + b is equal to : [JEE Main- 2016]
(A) –1 (B) 5 (C) 4 (D) 13
20. If A =
14–
3–2, then adj (3A2 + 12A) is equal to : [JEE Main - 2017]
(A)
7263
8451(B)
5184–
63–72(C)
5163–
84–72(D)
51 63
84 72
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21. Let A be a matrix such that A•1 2
0 3
is a scalar matrix and |3A| = 108. Then AA2 equals : [JEE Main - 2018]
(A) 4 32
0 36
(B) 36 0
32 4
(C) 4 0
32 36
(D) 36 32
0 4
22. Suppose A is any 3 × 3 non-singular matrix and (A – 3I) (A – 5I) = O, where I = I3 and O=O3. If A + A–1 =
4I, then + is equal to : [JEE Main - 2018]
(A) 8 (B) 7 (C) 13 (D) 12
23. Let A =
1 0 0
1 1 0
1 1 1
and B = AA20. Then the sum of the elements of the first column of B is [JEE Main - 2018]
(A) 211 (B) 210 (C) 231 (D) 251
EXERCISE-V
1. If matrix A =
bacacbcba
where a, b, c are real positive numbers, abc = 1 and AATA = I, then find the value of
a3 + b3 + c3 . [JEE 2003, Mains-2 out of 60]
2. If A =
2
2 and then =
(A) 3 (B) 2 (C) 5 (D) 0 [JEE 2004(Scr)]
3. If M is a 3 × 3 matrix, where MTM = I and det (M) = 1, then prove that det (M – I) = 0. [JEE 2004, 2 out of 60]
4. A =
cb1db101a
, B =
hgfcd011a
, U =
hgf
, V =
00a2
, X =
zyx
.
If AX = U has infinitely many solution, then prove that BX = V cannot have a unique solution. If further afd 0, then prove that BX = V has no solution. [JEE 2004, 4 out of 60]
5. A =
420110001
, I =
100010001
and AA–1 =
)dIcAA(6
1 2, then the value of c and d are
(A) –6, –11 (B) 6, 11 (C) –6, 11 (D) 6, – 11 [JEE 2005(Scr)]
6. If P =
2321
2123, A =
1011
and Q = PAPAPT and x = PTQ2005 P, then x is equal to
(A)
1020051
(B)
3200542005
6015320054
(C)
321
1324
1(D)
200532
3220054
1 [JEE 2005 (Screening)]
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SComprehension (3 questions) [JEE 2006, 5 marks each]
7.
123012001
A , U1, U2 and U3 are columns matrices satisfying. AU1 =
001
; AUAU2 =
032
, AU3 =
132
and
U is 3 × 3 matrix whose columns are U1, U2, U3 then answer the following questions
(a) The value of | U | is
(A) 3 (B) – 3 (C) 3/2 (D) 2
(b) The sum of elements of U–1 is
(A) – 1 (B) 0 (C) 1 (D) 3
(c) The value of
023
U023 is
(A) 5 (B) 5/2 (C) 4 (D) 3/2
8. Match the statements / Expression in Column-I with the statements / Expressions in Column-II and
indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in OMR.
Column-I Column-II
(A) The minimum value of 2x
4x2x2
is (P) 0
(B) Let A and B be 3 × 3 matrices of real numbers, (Q) 1
where A is symmetric, B is skew-symmetric, and
(A + B)(A – B) = (A – B)(A + B). If (AB)t = (–1)kAB, where (AB)t
is the transpose of the matrix AB, then the possible values of k are
(C) Let a = log3 log32. An integer k satisfying 1 < )3k( a2
< 2, must be (R) 2
less than
(D) If sin = cos , then the possible values of
2
1 are (S) 3
[JEE 2008, 6]
Paragraph for Question Nos. 9 to 11
Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are
1 and four of them are 0. [JEE-2009]
9. The number of matrices in A is
(A) 12 (B) 6 (C) 9 (D) 3
10. The number of matrices A in A for which the system of linear equations
0
0
1
z
y
x
A has a unique solution, is
(A) less than 4 (B) at least 4 but less than 7
(C) at least 7 but less than 10 (D) at least 10
11. The number of matrices A in A for which the system of linear equations
0
0
1
z
y
x
A is inconsistent, is
(A) 0 (B) more than 2 (C) 2 (D) 1
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12. The number of 3 × 3 matrices A whose entries are either 0 or 1 and for which the system
0
0
1
z
y
x
A has
exactly two distinct solutions, is(A) 0 (B) 29 – 1 (C) 168 (D) 2 [JEE-2010]
Paragraph for Questions 13 to 15Let P be an odd prime number and TP be the following set of 2 × 2 matrices :
}1–P....,2,1,0{c,b,a;
ac
baATP
13. The number of A in Tp such that A is either symmetric or skew-symmetric or both, and det(A) divisible by p is(A) (p – 1)2 (B) 2(p – 1) (C) (p – 1)2 + 1 (D) 2p – 1
14. The number of A in Tp such that the trace of A is not divisible by p but det (A) is divisible by p is[Note : The trace of a matrix is the sum of its diagonal entries.](A) (p – 1) (p2 – p + 1) (B) p3 – (p – 1)2 (C) (p – 1)2 (D) (p –1) (p2 – 2)
15. The number of A in Tp such that det (A) is not divisible by p is [JEE-2010](A) 2p2 (B) p3 – 5p (C) p3 – 3p (D) p3 – p2
Paragraph for question nos. 16 to 18
Let a, b and c be three real numbers satisfying [a b c]
737
728
791
= [0 0 0] .......(E) [JEE-2011]
16. If the point P(a, b, c), with reference to (E), lies on the plane 2x + y + z = 1, then the value of 7a + b + c is(A) 0 (B) 12 (C) 7 (D) 6
17. Let be a solution of x3 – 1 = 0 with Im() > 0. If a = 2 with b and c satisfying (E), then the value of
cba
313
is equal to
(A) –2 (B) 2 (C) 3 (D) –3
18. Let b = 6, with a and c satisfying (E). If and are the roots of the quadratic equation ax2 + bx + c = 0, then
is11
n
0n
(A) 6 (B) 7 (C) 6/7 (D)
19. Let M and N be two 3 × 3 non-singular skew symmetric matrices such that MN = NM. If PT denotes thetranspose of P, then M2N2 (MTN)–1 (MN–1)T is equal to [JEE-2011](A) M2 (B) – N2 (C) – M2 (D) MN
20. Let 1 be a cube root of unity and S be the set of all non-singular matrices of the form ,
1
c1
ba1
2
where
each of a, b, and c is either or 2. Then the number of distinct matrices in the set S is(A) 2 (B) 6 (C) 4 (D) 8 [JEE-2011]
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21. Let M be a 3 × 3 matrix satisfying [JEE-2011]
.
12
0
0
1
1
1
Mand,
1–
1
1
0
1–
1
M,
3
2
1–
0
1
0
M
Then the sum of the diagonal entries of M is :
22. Let P = [aij] be a 3 × 3 matrix and let Q = [bij] where bij = 2i+jaij for 1 i,j 3. If the determinant of P is 2, then
the determinant of the matrix Q is [JEE-2012]
(A) 210 (B) 211 (C) 212 (D) 213
23. If P is a 3 × 3 matrix such that PT = 2P + I , where PT is the transposes of P and I is the 3 × 3 identity matrix,
then there exists a column matrix X =
0
0
0
z
y
x
such that [JEE-2012]
(A) PX =
0
0
0
(B) PX = X (C) PX = 2X (D) PX = – X
24. If the adjoint of a 3 × 3 matrix P is
311
712
441
, then the possible value(s) of the determinant of P is (are)
(A) – 2 (B) – 1 (C) 1 (D) 2 [JEE-2012]
25. For 3 × 3 matrices M and N, which of the following statement(s) is (are) NOT correct?
(A) NT M N is symmetric or skew symmetric, according as M is symmetric or skew symmetric
(B) M N – N M is skew symmetric for all symmetric matrices M and N
(C) M N is symmeric for all symmetric matrices M and N
(D) (adj M) (adj N) = adj (M N) for all invertible matrices M and N [JEE Advanced - 2013]
26. Let M be a 2 × 2 symmetric matrix with integer entries. Then M is invertible if
(A) the first column of M is the transpose of the second row of M [JEE Advanced 2014]
(B) the second row of M is the transpose of the first column of M
(C) M is a diagonal matrix with non-zero entries in the main diagonal
(D) the product of entries in the main diagonal of M is not thesquare of an integer
27. Let M and N be two 3 × 3 matrices such that MN = NM. Further, if M N2 and M2 = N4, then
(A) determinant of (M2 + MN2) is 0 [JEE Advanced 2014]
(B) there is a 3 × 3 non-zero matrix v such that (M2 + MN2)U is the zero matrix
(C) determinant of (M2 + MN2) 1
(D) for a 3 × 3 matrix U, if (M2 + MN2)U euqals the zero matrix then U is the zero matrix
28. Let X and Y be two arbitrary, 3 × 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3, non-zero,
symmetric matrix. Then which of the following matrices is (are) skew symmetric? [JEE Advanced 2015]
(A) Y3Z4 – Z4Y3 (B) X44 + Y44 (C) X4Z3 – Z3X4 (D) X23 + Y23
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29. Let P =
053
02
213
, where R. Suppose Q = [qij] is a matrix such that PQ = kI, where k R, k 0
and I is the identity matrix of order 3. If q23 = –8
k and det(Q) =
2
k2
, then [JEE Advanced 2016]
(A) = 0, k = 8 (B) 4– k + 8 = 0(C) det(Padj(Q)) = 29 (D) det(Q adj(P)) = 213
30. Let
1416
014
001
P and I be the identity matrix of order 3. If Q = [qij] is a matrix such that P50 – Q = I, then
21
3231
q
qq equals [JEE Advanced 2016]
(A) 52 (B) 103 (C) 201 (D) 205
31. For a real number , if the system
1
1–
1
z
y
x
1
1
1
2
2
of linear equations, has infinitely many solutions,
then 1 + + 2 = [JEE Advanced 2017]
32. Which of the following is (are) NOT the square of a 3 × 3 matrix with real entries ? [JEE-Advanced-2017]
(A)
1 0 0
0 1 0
0 0 1
(B)
1 0 0
0 1 0
0 0 1
(C)
1 0 0
0 1 0
0 0 1
(D)
1 0 0
0 1 0
0 0 1
33. Let S be the set of all column matrices
3
2
1
b
b
b
such that b1, b2, b2 and the system of equations (in real
variables)
–x + 2y + 5z = b1
2x – 4y + 3z = b2
x – 2y + 2z = b3
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one
solution for each
3
2
1
b
b
b
S? [JEE Advanced 2018]
(A) x + 2y + 3z = b1, 4y + 5z = b2 and x + 2y + 6z = b3
(B) x + y + 3z = b1, 5x + 2y + 6z = b2 and –2x – y – 3z = b3
(C) –x + 2y – 5z = b1, 2x – 4y + 10z = b2 and x – 2y + 5z = b3
(D) x + 2y + 5z = b1, 2x + 3z = b2 and x + 4y – 5z = b3
34. Let P be a matrix of order 3 × 3 such that all the entries in P are from the set {–1, 0, 1}. Then, the maximum
possible value of the determinant of P is _________. [JEE Advanced 2018]
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35. Let M =
42
24
coscos1
sin–1–sin = I + M–1
where = () and = () are real numbers, and I is the 2 × 2 identity matrix. If
* is the minimum of the set {() : [0, 2)} and
* is the minimum of the set {() : [0, 2)},
then the value of * + * is : [JEE Advanced 2019]
(A) – 16
29(B) –
16
37(C)
16
17– (D)
16
31–
36. Let
0 1 a
M 1 2 3
3 b 1
and adj
–1 1 –1
M 8 –6 2
–5 3 –1
where a and b are real numbers. Which of the following options is/are correct? [JEE Advanced 2019]
(A) (adj M)–1 + adj M–1 = – M (B) If
1
M 2
3
, then – + = 3
(C) det(adj M2) = 81 (D) a + b = 3
37. Let x R and let [JEE Advanced 2019]
P =
1 1 1
0 2 2
0 0 3
, Q =
2 x x
0 4 0
x x 6
and R = PQP–1.
Then which of the following options is/are correct?(A) There exists a real number x such that PQ = QP
(B) det R = det
2 x x
0 4 0
x x 5
+ 8, for all x R
(C) For x = 0, if R
1
a
b
= 6
1
a
b
, then a + b = 5
(D) For x = 1, there exists a unit vector ˆ ˆ ˆi j k for which R
0
0
0
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S38. Let
P1 = I =
1 0 0
0 1 0
0 0 1
, P2 =
1 0 0
0 0 1
0 1 0
, P3 =
0 1 0
1 0 0
0 0 1
,
P4 =
0 1 0
0 0 1
1 0 0
, P5 =
0 0 1
1 0 0
0 1 0
, P6 =
0 0 1
0 1 0
1 0 0
and X = 6
Tk k
k 1
2 1 3
P 1 0 2 P
3 2 1
[JEE Advanced 2019]
where TkP denotes the transpose of the matrix Pk. Then which of the following options is/are correct?
(A) X – 30I is an invertible matrix (C) X is a symmetric matrix
(C) If X
1 1
1 1
1 1
, then = 30 (D) The sum of diagonal entries of X is 18
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ANSWER KEY
PROFICIENCY TEST-01
1. B 2. B 3. C 4. D 5. A 6. D 7. C
8. B 9. B 10. D 11. B 12. D 13. B 14. B
15. D
PROFICIENCY TEST-02
1. A 2. C 3. D 4. B 5. B 6. A 7. C
8. C 9. B 10. C 11. B 12. B 13. D 14. C
15. B
EXERCISE-I
1. 8 2. x = 2
3, y = 2 3. 5049 4. V =
11
1
0
6. 1
7. f (a) = 1/4, a = 1/2 9.
1401401
10.
22,
3
2,
3
24,
22,
3
2,
3
24,(3,3, –1)
11. 2 12. AB is neither symmetric nor skew symmetric
13. 4 14. n = 7 15. 650
EXERCISE-II
1. 225 2.
25321
13010
19417
3.
553
7744. 3/2
6. (i) x = 2, y = 1, z = 0 ; (ii) x = 2 + k, y = 1 2k, z = k where k R ;
(iii) inconsistent, hence no solution
7. 1 8.
4270
2548
19
19.
k
mn
mkn
1
10. (i) X =
b21a22
ba for a, b R ; (ii) X does not exist ;
(iii) X =
c3c
a3aa, c R and 3a + c 0; 3b + d 0
11. X =
dc
d2c2, where c, d R – {0}, NO
12. (i) a – 3 , b R ; (ii) a = – 3 and b 1/3 ; (iii) a = –3 , b = 1/3
13. (a) X=
22
533
, (b) X =
21
21, (c) no solution 15. (A) Q; (B) S; (C) P; (D) P
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MA
TR
ICE
S
EXERCISE-III
1. D 2. D 3. D 4. D 5. B 6. C 7. B8. D 9. D 10. C
EXERCISE-IV1. B 2. B 3. B 4. D 5. A 6. B 7. D8. C 9. D 10. B 11. A 12. D 13. C 14. D15. C 16. C 17. C 18. A 19. B 20. D 21. D
22. A 23. C
EXERCISE-V
1. 4 2. A 5. C 6. A 7. (a) A, (b) B, (c) A
8. (A) R (B) Q,S (C) R,S (D) P,R 9. A 10. B 11. B 12. A
13. D 14. C 15. D 16. D 17. A 18. B 19. Bonus
20. A 21. 9 22. D 23. D 24. A, D 25. C, D 26. C, D
27. A,B 28. C,D 29. B, C 30. B 31. 1 32. A,C 33. A, D
34. 4 35. A 36. ABD 37. BC 38. BCD
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