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MATH1131 Mathematics 1A Algebra
UNSW Sydney Semester 1, 2017
Maike Massierer
Chapter 5: Matrices
Lecture 21: Operations on matrices
DefinitionA matrix A is an array of elements aij in rows and columns:
A = (aij) =
a11 a12 · · · a1na21 a22 · · · a2n...
......
am1 am2 · · · amn
DefinitionA matrix A is an array of elements aij in rows and columns:
A = (aij) =
a11 a12 · · · a1na21 a22 · · · a2n...
......
am1 am2 · · · amn
Here, A has m rows and n columns, so
A is an m× n matrix
the size of A is m× n
if m = n, then A is square.
DefinitionA matrix A is an array of elements aij in rows and columns:
A = (aij) =
a11 a12 · · · a1na21 a22 · · · a2n...
......
am1 am2 · · · amn
Here, A has m rows and n columns, so
A is an m× n matrix
the size of A is m× n
if m = n, then A is square.
Mmn is the set of all m× n matrices.
DefinitionA matrix A is an array of elements aij in rows and columns:
A = (aij) =
a11 a12 · · · a1na21 a22 · · · a2n...
......
am1 am2 · · · amn
Here, A has m rows and n columns, so
A is an m× n matrix
the size of A is m× n
if m = n, then A is square.
Mmn is the set of all m× n matrices.Each row of A is a row vector, and each column of A is a column vector.
DefinitionA matrix A is an array of elements aij in rows and columns:
A = (aij) =
a11 a12 · · · a1na21 a22 · · · a2n...
......
am1 am2 · · · amn
Here, A has m rows and n columns, so
A is an m× n matrix
the size of A is m× n
if m = n, then A is square.
Mmn is the set of all m× n matrices.Each row of A is a row vector, and each column of A is a column vector.The elements of A are also called entries.
DefinitionA matrix A is an array of elements aij in rows and columns:
A = (aij) =
a11 a12 · · · a1na21 a22 · · · a2n...
......
am1 am2 · · · amn
Here, A has m rows and n columns, so
A is an m× n matrix
the size of A is m× n
if m = n, then A is square.
Mmn is the set of all m× n matrices.Each row of A is a row vector, and each column of A is a column vector.The elements of A are also called entries.The entry aij in row i and column j of A is also denoted by [A]ij.
ExampleConsider the matrix
A =
0 2 1 −13 1 4 0−1 0 −2 2
ExampleConsider the matrix
A =
0 2 1 −13 1 4 0−1 0 −2 2
A has 3 rows and 4 columns.
ExampleConsider the matrix
A =
0 2 1 −13 1 4 0−1 0 −2 2
A has 3 rows and 4 columns.Therefore, A has size 3× 4 and is an element of M34.
ExampleConsider the matrix
A =
0 2 1 −13 1 4 0−1 0 −2 2
A has 3 rows and 4 columns.Therefore, A has size 3× 4 and is an element of M34. A is not square.
ExampleConsider the matrix
A =
0 2 1 −13 1 4 0−1 0 −2 2
A has 3 rows and 4 columns.Therefore, A has size 3× 4 and is an element of M34. A is not square.Some of the entries of A are
[A]12 = 2 [A]24 = 0 [A]31 = −1
DefinitionLet A and B be m× n matrices and let λ be a number.
DefinitionLet A and B be m× n matrices and let λ be a number.
Addition A+B is anm×n matrix with [A+B]ij = [A]ij+[B]ij
DefinitionLet A and B be m× n matrices and let λ be a number.
Addition A+B is anm×n matrix with [A+B]ij = [A]ij+[B]ij
Subtraction A−B is anm×n matrix with [A−B]ij = [A]ij−[B]ij
DefinitionLet A and B be m× n matrices and let λ be a number.
Addition A+B is anm×n matrix with [A+B]ij = [A]ij+[B]ij
Subtraction A−B is anm×n matrix with [A−B]ij = [A]ij−[B]ij
Scalar Multiplication λA is an m× n matrix with [λA]ij = λ[A]ij
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
We have
A+ B =
A− B =
2A =
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
We have
A+ B =
(
0 2 13 1 4
)
+
(
5 1 33 2 −1
)
A− B =
2A =
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
We have
A+ B =
(
0 2 13 1 4
)
+
(
5 1 33 2 −1
)
=
(
5 3 46 3 3
)
A− B =
2A =
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
We have
A+ B =
(
0 2 13 1 4
)
+
(
5 1 33 2 −1
)
=
(
5 3 46 3 3
)
A− B =
(
0 2 13 1 4
)
−
(
5 1 33 2 −1
)
2A =
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
We have
A+ B =
(
0 2 13 1 4
)
+
(
5 1 33 2 −1
)
=
(
5 3 46 3 3
)
A− B =
(
0 2 13 1 4
)
−
(
5 1 33 2 −1
)
=
(
−5 1 −20 −1 5
)
2A =
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
We have
A+ B =
(
0 2 13 1 4
)
+
(
5 1 33 2 −1
)
=
(
5 3 46 3 3
)
A− B =
(
0 2 13 1 4
)
−
(
5 1 33 2 −1
)
=
(
−5 1 −20 −1 5
)
2A = 2
(
0 2 13 1 4
)
ExampleConsider the 2× 3 matrices
A =
(
0 2 13 1 4
)
and B =
(
5 1 33 2 −1
)
We have
A+ B =
(
0 2 13 1 4
)
+
(
5 1 33 2 −1
)
=
(
5 3 46 3 3
)
A− B =
(
0 2 13 1 4
)
−
(
5 1 33 2 −1
)
=
(
−5 1 −20 −1 5
)
2A = 2
(
0 2 13 1 4
)
=
(
0 4 26 2 8
)
ExerciseCalculate the following matrices if possible:(
3 21 3
)
+
(
5 1−1 1
)
= 3
(
1 20 3
)
=
(
4 2)
−(
2 1)
=
(
3 12 3
)
+
(
5−1
)
=
ExerciseCalculate the following matrices if possible:(
3 21 3
)
+
(
5 1−1 1
)
=
(
8 30 4
)
3
(
1 20 3
)
=
(
4 2)
−(
2 1)
=
(
3 12 3
)
+
(
5−1
)
=
ExerciseCalculate the following matrices if possible:(
3 21 3
)
+
(
5 1−1 1
)
=
(
8 30 4
)
3
(
1 20 3
)
=
(
4 2)
−(
2 1)
=(
2 1)
(
3 12 3
)
+
(
5−1
)
=
ExerciseCalculate the following matrices if possible:(
3 21 3
)
+
(
5 1−1 1
)
=
(
8 30 4
)
3
(
1 20 3
)
=
(
3 60 9
)
(
4 2)
−(
2 1)
=(
2 1)
(
3 12 3
)
+
(
5−1
)
=
ExerciseCalculate the following matrices if possible:(
3 21 3
)
+
(
5 1−1 1
)
=
(
8 30 4
)
3
(
1 20 3
)
=
(
3 60 9
)
(
4 2)
−(
2 1)
=(
2 1)
(
3 12 3
)
+
(
5−1
)
= not defined!
DefinitionA zero matrix 0 is a matrix with only zero entries.
DefinitionA zero matrix 0 is a matrix with only zero entries.As with vectors, we often talk about the zero matrix 0 if its size is given.
DefinitionA zero matrix 0 is a matrix with only zero entries.As with vectors, we often talk about the zero matrix 0 if its size is given.
ExampleThe following matrices are zero matrices:
0 =
(
0 00 0
)
∈ M22 0 =
(
0 0 00 0 0
)
∈ M23
TheoremFor all matrices A,B,C ∈ Mmn and scalars λ, µ, the following hold:
Closure under Addition A+B ∈ Mmn
Associative Law of Addition (A+B) + C = A+ (B + C)Commutative Law of Addition A+B = B + A
Existence of Zero Some 0 ∈ Mmn satisfies A+0 = A for all A ∈ Mmn
Existence of Negative Some element−A ∈ Mmn satisfies A+(−A) = 0
Closure under Scalar Multiplication λA ∈ Mmn
Associative Law of Scalar Multiplication λ(µA) = (λµ)AMultiplication by identity 1A = A
Scalar Distributive Law (λ+ µ)A = λA+ µA
Vector Distributive Law λ(A+B) = λA+ λB
ExampleProve the Commutative Law of Addition for Mmn : A+B = B + A
ExampleProve the Commutative Law of Addition for Mmn : A+B = B + A
ProofCompare each entry of A+ B and B + A:
[A+B]ij
ExampleProve the Commutative Law of Addition for Mmn : A+B = B + A
ProofCompare each entry of A+ B and B + A:
[A+B]ij = [A]ij + [B]ij
ExampleProve the Commutative Law of Addition for Mmn : A+B = B + A
ProofCompare each entry of A+ B and B + A:
[A+B]ij = [A]ij + [B]ij = [B]ij + [A]ij
ExampleProve the Commutative Law of Addition for Mmn : A+B = B + A
ProofCompare each entry of A+ B and B + A:
[A+B]ij = [A]ij + [B]ij = [B]ij + [A]ij = [B + A]ij .
ExampleProve the Commutative Law of Addition for Mmn : A+B = B + A
ProofCompare each entry of A+ B and B + A:
[A+B]ij = [A]ij + [B]ij = [B]ij + [A]ij = [B + A]ij .
Since the two matrices have the same entries, they are equal. ✷
ExerciseFill in the missing parts of the following proof ofthe Associative Law of Addition for Mmn : (A+B)+C = A+(B+C)
ProofCompare each entry of (A+B) + C and A+ (B + C):
[(A+ B) + C]ij =
= ([A]ij + [B]ij) + [C]ij
=
= [A]ij + ([B + C]ij)
=
Since the two matrices have the same entries, they are equal. ✷
ExerciseFill in the missing parts of the following proof ofthe Associative Law of Addition for Mmn : (A+B)+C = A+(B+C)
ProofCompare each entry of (A+B) + C and A+ (B + C):
[(A+ B) + C]ij = [A+ B]ij + [C]ij
= ([A]ij + [B]ij) + [C]ij
=
= [A]ij + ([B + C]ij)
=
Since the two matrices have the same entries, they are equal. ✷
ExerciseFill in the missing parts of the following proof ofthe Associative Law of Addition for Mmn : (A+B)+C = A+(B+C)
ProofCompare each entry of (A+B) + C and A+ (B + C):
[(A+ B) + C]ij = [A+ B]ij + [C]ij
= ([A]ij + [B]ij) + [C]ij
= [A]ij + ([B]ij + [C]ij)
= [A]ij + ([B + C]ij)
=
Since the two matrices have the same entries, they are equal. ✷
ExerciseFill in the missing parts of the following proof ofthe Associative Law of Addition for Mmn : (A+B)+C = A+(B+C)
ProofCompare each entry of (A+B) + C and A+ (B + C):
[(A+ B) + C]ij = [A+ B]ij + [C]ij
= ([A]ij + [B]ij) + [C]ij
= [A]ij + ([B]ij + [C]ij)
= [A]ij + ([B + C]ij)
= [A+ (B + C)]ij
Since the two matrices have the same entries, they are equal. ✷
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =n
∑
k=1
aikbkj
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =n
∑
k=1
aikbkj
Note that the number of columns of A (= n) has to be the same as thenumber of rows of B (= n). If this is not true, then AB is not defined.
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =n
∑
k=1
aikbkj
Note that the number of columns of A (= n) has to be the same as thenumber of rows of B (= n). If this is not true, then AB is not defined.
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =n
∑
k=1
aikbkj
Note that the number of columns of A (= n) has to be the same as thenumber of rows of B (= n). If this is not true, then AB is not defined.
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =n
∑
k=1
aikbkj
Note that the number of columns of A (= n) has to be the same as thenumber of rows of B (= n). If this is not true, then AB is not defined.
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =n
∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
( )
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
( )
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
(
3(−1) + 4× 5 + 2(−4))
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
(
3(−1) + 4× 5 + 2(−4))
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
(
3(−1) + 4× 5 + 2(−4))
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
(
3(−1) + 4× 5 + 2(−4)0(−1) + 2× 5 + 1(−4)
)
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
(
3(−1) + 4× 5 + 2(−4)0(−1) + 2× 5 + 1(−4)
)
DefinitionLet A = (aik) be an m× n matrix and let B = (bkj) be an n× p matrix.
Matrix Multiplication AB is an m× p matrix with entries
[AB]ij = ai1b1j + · · · + ainbnj =
n∑
k=1
aikbkj
ExampleConsider the matrices
A =
(
3 4 20 2 1
)
B =
−15−4
Now, A is a 2× 3 matrix, and B is a 3× 1 matrix.Therefore, AB is a well-defined 2× 1 matrix:
AB =
(
3 4 20 2 1
)
−15−4
=
(
3(−1) + 4× 5 + 2(−4)0(−1) + 2× 5 + 1(−4)
)
=
(
96
)
ExerciseCalculate
(
1 3)
(
−12
)
=
(
13
)
(
−1 2)
=
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 43× 2 3× 21× 1 1× 22× 3 3× 2
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 4 Yes3× 2 3× 21× 1 1× 22× 3 3× 2
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 4 Yes 3× 43× 2 3× 21× 1 1× 22× 3 3× 2
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 4 Yes 3× 43× 2 3× 2 No1× 1 1× 22× 3 3× 2
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 4 Yes 3× 43× 2 3× 2 No1× 1 1× 2 Yes2× 3 3× 2
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 4 Yes 3× 43× 2 3× 2 No1× 1 1× 2 Yes 1× 22× 3 3× 2
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 4 Yes 3× 43× 2 3× 2 No1× 1 1× 2 Yes 1× 22× 3 3× 2 Yes
ExerciseCalculate
(
1 3)
(
−12
)
=(
1(−1) + 3× 2)
=(
5)
(
13
)
(
−1 2)
=
(
1(−1) 1× 23(−1) 3× 2
)
=
(
−1 2−3 6
)
ExerciseConsider two matrices A and B.
Size of A Size of B Does AB exist? Size of AB (if it exists)3× 2 2× 4 Yes 3× 43× 2 3× 2 No1× 1 1× 2 Yes 1× 22× 3 3× 2 Yes 2× 2
DefinitionThe diagonal of squarematrix A = (aij) ∈ Mnn is the entries a11, . . . , ann.
DefinitionThe diagonal of squarematrix A = (aij) ∈ Mnn is the entries a11, . . . , ann.An identity matrix I is a square matrix with
all diagonal entries equal to 1 andall other entries equal to 0.
DefinitionThe diagonal of squarematrix A = (aij) ∈ Mnn is the entries a11, . . . , ann.An identity matrix I is a square matrix with
all diagonal entries equal to 1 andall other entries equal to 0.
As for 0, we often talk about the identity matrix I if its size is given.
DefinitionThe diagonal of squarematrix A = (aij) ∈ Mnn is the entries a11, . . . , ann.An identity matrix I is a square matrix with
all diagonal entries equal to 1 andall other entries equal to 0.
As for 0, we often talk about the identity matrix I if its size is given.
ExampleThe following matrices are identity matrices:
I =(
1)
∈ M11 I =
(
1 00 1
)
∈ M22 I =
1 0 00 1 00 0 1
∈ M33
DefinitionThe diagonal of squarematrix A = (aij) ∈ Mnn is the entries a11, . . . , ann.An identity matrix I is a square matrix with
all diagonal entries equal to 1 andall other entries equal to 0.
As for 0, we often talk about the identity matrix I if its size is given.
ExampleThe following matrices are identity matrices:
I =(
1)
∈ M11 I =
(
1 00 1
)
∈ M22 I =
1 0 00 1 00 0 1
∈ M33
LemmaLet A be an m×n matrix and Im ∈ Mmm, In ∈ Mnn be identity matrices.
DefinitionThe diagonal of squarematrix A = (aij) ∈ Mnn is the entries a11, . . . , ann.An identity matrix I is a square matrix with
all diagonal entries equal to 1 andall other entries equal to 0.
As for 0, we often talk about the identity matrix I if its size is given.
ExampleThe following matrices are identity matrices:
I =(
1)
∈ M11 I =
(
1 00 1
)
∈ M22 I =
1 0 00 1 00 0 1
∈ M33
LemmaLet A be an m×n matrix and Im ∈ Mmm, In ∈ Mnn be identity matrices.Then ImA = AIn = A.
LemmaLet A be an m×n matrix and Im ∈ Mmm, In ∈ Mnn be identity matrices.Then ImA = AIn = A.
LemmaLet A be an m×n matrix and Im ∈ Mmm, In ∈ Mnn be identity matrices.Then ImA = AIn = A.
LemmaFor all matrices A ∈ Mmn and B ∈ Mnp and scalars λ,
λ(AB) = (λA)B = A(λB).
LemmaLet A be an m×n matrix and Im ∈ Mmm, In ∈ Mnn be identity matrices.Then ImA = AIn = A.
LemmaFor all matrices A ∈ Mmn and B ∈ Mnp and scalars λ,
λ(AB) = (λA)B = A(λB).
TheoremFor all matrices A,B,C, we have that (if the products exist!):
Associative Law of Matrix Multiplication (AB)C = A(BC)Left Distributive Law (A+B)C = AC +BC
Right Distributive Law A(B + C) = AB + AC
TheoremFor all matrices A,B,C, we have that (if the products exist!):
Associative Law of Matrix Multiplication (AB)C = A(BC)Left Distributive Law (A+B)C = AC +BC
Right Distributive Law A(B + C) = AB + AC
TheoremFor all matrices A,B,C, we have that (if the products exist!):
Associative Law of Matrix Multiplication (AB)C = A(BC)Left Distributive Law (A+B)C = AC +BC
Right Distributive Law A(B + C) = AB + AC
RemarkMatrices do not generally multiply commutatively!That is, AB = BA is not generally true.
TheoremFor all matrices A,B,C, we have that (if the products exist!):
Associative Law of Matrix Multiplication (AB)C = A(BC)Left Distributive Law (A+B)C = AC +BC
Right Distributive Law A(B + C) = AB + AC
RemarkMatrices do not generally multiply commutatively!That is, AB = BA is not generally true.
Example(
1 00 0
)(
0 10 0
)
=
(
0 10 0
)
6=
(
0 00 0
)
=
(
0 10 0
)(
1 00 0
)
ExerciseExpand the expression
(A+B)2 =
ExerciseExpand the expression
(A+B)2 = (A+B)(A+B)
= A(A+B) +B(A+B)
= A2 + AB + BA+ B2
This might not equal A2 + 2AB +B2 !
DefinitionLet A = (aij) be an m× n matrix.
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
ExampleThe transpose of
A =
(
3 −4 02 0 1
)
is AT =
3 2−4 00 1
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
ExampleThe transpose of
A =
(
3 −4 02 0 1
)
is AT =
3 2−4 00 1
ExampleIf v is a column (row) vector, then v
T is a row (column) vector;
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
ExampleThe transpose of
A =
(
3 −4 02 0 1
)
is AT =
3 2−4 00 1
ExampleIf v is a column (row) vector, then v
T is a row (column) vector;Let u,v ∈ Mn1 be column vectors; then u
Tv is their scalar product.
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
ExampleThe transpose of
A =
(
3 −4 02 0 1
)
is AT =
3 2−4 00 1
ExampleIf v is a column (row) vector, then v
T is a row (column) vector;Let u,v ∈ Mn1 be column vectors; then u
Tv is their scalar product.
Here, we treat the 1× 1 matrix uTv as a scalar.
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
ExampleThe transpose of
A =
(
3 −4 02 0 1
)
is AT =
3 2−4 00 1
ExampleIf v is a column (row) vector, then v
T is a row (column) vector;Let u,v ∈ Mn1 be column vectors; then u
Tv is their scalar product.
Here, we treat the 1× 1 matrix uTv as a scalar.
As a special case, vTv = |v|2;
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
ExampleThe transpose of
A =
(
3 −4 02 0 1
)
is AT =
3 2−4 00 1
ExampleIf v is a column (row) vector, then v
T is a row (column) vector;Let u,v ∈ Mn1 be column vectors; then u
Tv is their scalar product.
Here, we treat the 1× 1 matrix uTv as a scalar.
As a special case, vTv = |v|2; for instance,
if v =
(
34
)
then vTv =
(
3 4)
(
34
)
= 32 + 42 = 25 = |v|2
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
DefinitionLet A = (aij) be an m× n matrix.
Transposition The transpose AT is the n×m matrix with [AT ]ij = [A]ji
If AT = A, then A is symmetric.
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
Example
Consider the matrix A =
(
1 20 2
)
.
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
Example
Consider the matrix A =
(
1 20 2
)
.
Note that
(AT )T =
((
1 20 2
)T)T
=
(
1 02 2
)T
=
(
1 20 2
)
= A
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
Example
Consider the matrices A =
(
1 20 2
)
and C =
(
0 11 0
)
.
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
Example
Consider the matrices A =
(
1 20 2
)
and C =
(
0 11 0
)
.
Note that
(AC)T =
((
1 20 2
)(
0 11 0
))T
=
(
2 12 0
)T
=
(
2 21 0
)
TheoremFor all matrices A,B ∈ Mmn and C ∈ Mnp and scalars λ, µ,
(AT )T = A
(λA+ µB)T = λAT + µBT
(AC)T = CTAT
Example
Consider the matrices A =
(
1 20 2
)
and C =
(
0 11 0
)
.
Note that
(AC)T =
((
1 20 2
)(
0 11 0
))T
=
(
2 12 0
)T
=
(
2 21 0
)
CTAT =
(
0 11 0
)T (
1 20 2
)T
=
(
0 11 0
)(
1 02 2
)
=
(
2 21 0
)
= (AC)T