TOPIC 4 MATRICES

MATRICES

4.0 INTRODUCTION

Matrices are sets of numbers that are arranged in rectangular forms. It is a rectangular

array of numbers. These numbers are arranged inside a round or square bracket. Look at the

examples shown below.

It is important to study about the fundamentals of matrices first and get a good introduction to how

to apply simple algebra operations on matrices. This can help in solving engineering problems.

For example, you can use matrices to solve systems of linear simultaneous equations.

4.1 FUNDAMENTALS OF MATRIX

The array of numbers inside a matrix is called the elements of the matrix. These

numbers are arranged in rows and columns.

Rows are the horizontally arranged elements of the matrixFor example, the shaded region in the matrix below is the second row of the matrix.

Columns are the vertically arranged elements of the matrix.For example, the shaded region in the matrix below is the second column of the matrix.

Note: It is common practice to use capital letters like A to represent a matrix, and small letters to

represent the elements.

4.2 SIZE OF A MATRIX

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The size of a matrix is the number of rows and columns that it has. If a matrix has 3

rows and 4 columns, then its size is 3 x 4. Let’s look at the following matrix.

A =

How many rows and columns do you see? Do you agree that the size of matrix A is 3 x 4?

Example 1:State the size of the following matrix.

Solution:There are 4 rows and 3 columns. Therefore, the size of this matrix is 4 x 3.

For a matrix A of size 3 x 4, you can use the notation A34 to represent the matrix. In general, any

matrix can be represented by the notation matrix Aik with i = 1, 2, 3, …., and k = 1, 2, 3, ……

The first subscript, i, represents the rows and the second subscript, j, represents the columns.

ACTIVITY 4a

1. State the size of each of the following matrices:

a. b. c. d.

2. Referring to matrix B = , state the element at:

a. b23

b. b21

c. b31

4.3 TYPES OF MATRIX

4.3.1 Square Matrix

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A square matrix is a matrix where the number of rows is equal to the number of

columns. The following examples are square matrices.

4.3.2 Diagonal Matrix

If all the elements of a square matrix consist of zeros except the diagonal, then this

matrix is called a diagonal matrix. The following examples are diagonal matrices.

4.3.3 Identity Matrix

If all the elements of a diagonal matrix consist of the value 1, then the matrix is

an identity matrix. The following examples are identity matrices.

I = I = I =

An identity matrix is special because when you multiply a matrix with it or when

you multiply it with a matrix, the matrix does not change. For examples:

AI = IA = A, IB = BI = B

4.3.4 Transpose of a Matrix

When you interchange the rows of a matrix with its columns, you would have

converted a matrix Amn to another matrix Anm. In other words, a matrix of size m x n will

now be of size n x m. This new matrix is called the transpose of a matrix. The symbol for a

transpose of a matrix A is AT. Let’s look at the following example.

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If A = , then AT = .

If A = , then AT = .

The transpose of a transpose is the original matrix. (AT)T = A

Some important properties relating to transpose are:

(AB)T = BTAT

(ABC…Z)T = ZT…..BTAT

(A + B)T = AT + BT

4.3.5 Symmetric Matrix

If the transpose of a matrix is the same as the original matrix, then it is called a

symmetric matrix. Therefore, if A = AT, then A is a symmetric matrix. The following

examples are symmetric matrices.

Activity 4b:

1. Determine the types of the following matrices:

A = B = C = D =

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2. Determine whether the following matrices are symmetric or not.

a) b) c) d)

4.4 MATRIX ADDITION AND SUBTRACTION

Matrix addition / subtraction can only be performed on matrices that have the same size.

The result of a matrix addition / subtraction is a new matrix that is of the same size. All we need

to do is to match the elements that are at the same position in their matrices.

Example 2:

Given that and

Determine A + B and A – B.

Solution:

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4.5 MATRIX MULTIPLICATION

In order to be able to multiply two matrices AB, we have to ensure that the number of

columns in matrix A is the same as the number of rows in matrix B. That means we can multiply

matrix Amn with matrix Bnk because matrix A has n columns and matrix B has n rows too. The

result is a new matrix that has m rows and k columns.

Example 3:

Find the multiplication of and .

Solution:

Example 4:Find the product of matrix A and B, given that

A = and B = .

Solution:.

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ACTIVITY 4c

1. Based on the following matrices,

A = B = C = D =

Determine:

a. A + Bb. A – Cc. D + (B – A)

d. B + C

2. Given that A = and B = , Find AB and BA.

3. If P = , Q = , R = and S = . Find the

product of: a) PQ b) P2 c) QI d) QR e) RS f) SQ

4. Given A = , B = and C = . Find the value for:

a) ( A + B )T b) ( AB )T c) CT BT

5. Find the values of x and y for the followings:

a) + =

b) =

4.6 DETERMINANT OF A MATRIX

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The determinant of a square matrix is a special number that can be calculated from the

matrix. It is used to represent the real-value of the matrix which can be used to solve simple

algebra problems later on. The symbol for the determinant of matrix A is det(A) or A.

For a matrix of size 2 x 2, the method to find the determinant is:

If A = ,

then, det(A) = A = = (ad – bc).

For a matrix of size 3 x 3, the method to find the determinant is:

If A = ,

then A=

therefore, A=

Example 5:

If A = , determine det(A).

Solution:

Example 6:

Determine the determinant of matrix

Solution:Method 1 : Matrix minor (Matriks Minor)

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Method 2 : Cross Multiplication (Darab silang)

ACTIVITY 4d

1.Determine the determinants for the following 2x2 matrices:

a) b) c)

2.Given that A = , B = and C =

Determine:

a) A b) B c) C

4.7 MINOR OF A MATRIX

The Minor of a matrix is a new matrix where all the elements are determinants. Each

determinant is calculated by removing a row and a column from the original matrix. For example,

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in order to determine the element at position ij, you will have to remove row i and column j from

the matrix. Next, you calculate the determinant of what is left.

If A = , then Minor of A =

Where:

by removing row 1 and column 1 from A

by removing row 1 and column 2 from A

by removing row 2 and column 3 from A

and so on…

Example 7:

If A = , determine Minor of A.

Solution:The elements are:

= 9(8) – 2(6) = 60 = 7(8) – 2(4) = 48

= 7(6) – 9(4) = 6 = 3(8) – 5(6) = - 6

= 1(8) – 5(4) = -12 = 1(6) – 3(4) = -6

= 3(2) – 5(9) = -30 = 1(2) – 5(7) = -33

= 1(9) – 3(7) = -12

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Therefore, Minor of A =

ACTIVITY 4eFind the minor for the following matrices:

i) A = ii) B =

4.8 COFACTOR OF A MATRIX

Once you have found the Minor of a matrix, you can easily determine the Cofactor of the

matrix. All the hard work is already done when you determine the Minor of a matrix. All you need

to do now is to multiply each element of the Minor of the matrix with a factor and the

Cofactor is done.

Let’s look at the following descriptions:

If A = and Minor A =

Then, Cofactor of matrix A = where .

Therefore,

Cofactor of matrix A =

Example 8:

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If A = , determine the Cofactor of matrix A.

Solution:

First, the Minor of matrix A =

Next, multiply each element by its factor

Therefore, you get the

Cofactor of matrix A =

=

ACTIVITY 4fFind the cofactor for the following matrices:

i) A = ii) B =

4.9 ADJOINT OF MATRIX

For a square matrix A with n x n, you can find the adjoint of a matrix when transposing

the cofactors of a matrix A. In this case, for matrix A,

Adjoint of a matrix A, written as Adj(A) = KT where K are the cofactor for A.

Then, if A = and Minor A =

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And Cofactor matrix A =

Then adjoint matrix A, Adj(A) = KT =

Example 9:

If A = , determine the adjoint for matrix A.

Solution:Inputs from Example 9 and Example 10, you will find minor for A =

And matrix of cofactor from A =

Then, adjoint of matrix A, Adj(A) = AT =

AKTIVITY 4gFind the adjoin matrix for the following matrices:

i) A = ii) B =

4.10 INVERSE MATRIX

If A is a square matrix, A-1 is called the inverse matrix of A. Then AA-1 = I (Identity matrix).

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ACTIVITY 4hCalculate the inverse matrix for each of the matrix below:

i) A = ii) B =

4.11 SYSTEMS OF LINEAR EQUATIONS

You have understood the different types of matrices and their operations. You have also

learned to determine the inverse of a matrix. Using the inverse of a matrix, you can solve

simultaneous equations using Cramer Rule and the Inverse method.

4.11.1 THE INVERSE METHOD

Consider the system of equation:

You have to write it in a matrix form, Ac = b.

AdjAA

A 11

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=

Where matrix A = , matrix c = and matrix b = . Therefore c

= A-1b.

To determine c = , you have to multiply the inverse of A, A-1 to b.

Example 10:Solve the linear system:

x + 3y + 3z = 4

2x –3y –2z = 2

3x + y + 2z = 5

Solution:

Rewrite in the form of matrix, Ac = b:

213232

331 =

Determine A-1 for matrix A,

Determinant for A, = -1 and Minor of A =

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Next, we find the cofactor of A which is

Now the adjoin of A = AT is

Finally, the inverse of A is

Therefore =

=

x = 7, y = 14, z = -15

4.11.2 CRAMER’S RULE

Another method of solving systems of linear equations is using Cramer’s Rule where you have to

calculate the determinants of the matrices involved.

For a matrix equation, =

Let A = .

You get A1, by substituting into column 1 of matrix A.

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Therefore A1 =

Using the same method for A2 = and A3 =

According to Cramer’s Rule:

= , = , and = ,

Example 11:

Solve for x, y and z

5x - y + 7z = 4

6x - 2y + 9z= 5

2x + 8y –4z= 8

Solution:Writing in a matrix form:

Let A = .

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A1 = A2 = and A3 =

=2 = 44 = -26 and = -34

= = = 22

= = = 13

= = = -17

ACTIVITY 4i

1. Solve the following system of linear equations using the inverse method.

a) 2x + y + z = 8

5x – 3y + 2z = 3

7x + y + 3z = 20

b) 3x + 2y + 4z = 3

x + y + z = 2

2x – y + 3z = -3

2. Solve the following system of linear equations using the Cramer’s Rule.

a) X1 + 2x2 – X3 = 4

3X1 – 4X2 – 2X3 = 2

5X1 + 3X2 + 5X3 = -1

b) 4a – 5b + 6c = 3

8a – 7b – 3c = 9

7a – 8b + 9c = 6

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3. Solve the following system of linear equations using the inverse method and Cramer’s Rule.

a) x + y + 2z = 1

2x + 3y + 6z = 1

3x + 2y – 4z = 2

b) 3x + 2y – z = 10

7x – y + 6z = 8

3x + 2z – 5 = 0

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