Matrix - Operations on Matrices and their Properties - Unit 3

Operations on Matrices and Their Properties

Normally, we have four main operations that we do in mathematics: addition, subtraction, multiplication and division. Here we will discuss addition, subtraction, transpose and multiplication of matrices, along with some properties.

Previously, we discussed about:

Introduction to Matrix

Notation of Matrix

Order or size of Matrix

Special Types of Matrices

Today, our contents are as follows:

Addition and Subtraction of Matrices

Properties of Matrix Addition

Transpose of A Matrix

Multiplication of Matrices

Properties of Matrix Multiplication

Addition and Subtraction of Matrices:

To know about addition and subtraction of Matrices, you will need to understand the notation of elements and size of a matrix.

Remember these things:

Two matrices can be added or subtracted only if they are of the same order.
Only the corresponding elements of each matrices can be added or subtracted.
The result (either sum or difference) is the matrix of same order.

Addition of Matrices:

Let A and B be two matrices of order 2x2, each.

We can only add:

a₁₁ + b₁₁

a₁₂ + b₁₂

a₂₁ + b₂₁

a₂₂ + b₂₂

Here is an example of addition of matrices:

Addition of 2x2 matrices

Subtraction of Matrices:

After subtracting two or more matrices, you will get the difference as result. Note: the result of matrices B-A is not equal to A-B.

Let A and B be two matrices of order 2x2, each. We can only subtract (A-B) as:

a11 - b11

a12 - b12

a₂₁ - b₂₁

a₂₂ - b₂₂

And, (B-A) as:

b₁₁ - a₁₁

b₁₂ - a₁₂

b₂₁ - a₂₁

b₂₂ - a₂₂

Take a look at this example:

Subtraction of 2x2 Matrices

Properties of Matrix Addition:

Closure Property:

When two matrices of same size are added, the result is also a matrix of the same size.

i.e A_1x2 + B_1x2 = (A+B)_1x2 [Proof]

Commutative Property:

When two matrices of same size are added, the result is always the same.

i.e. A+ B = B+A

Associative Property:

When three matrices of same size are added then, the result is always the same.

i.e. (A+B)+C = A +(B+C)

Identity Property:

Let us consider a matrix A of order lxm and a null or zero matrix O of the same order lxm. On adding these matrices, the result is always equal to matrix A.

i.e A+O = A = O+A

Additive Inverse:

Let a matrix be A. When you add matrix -A to A, you receive a Null Matrix.

So, for every matrices, there exists a negative matrix which when added results a null matrix.

i.e. A +(-A) = (-A) +A = O

Hence, when we multiply any matrix by a '-' sign, we receive the additive inverse of the particular matrix.

Transpose of a Matrix:

In general understanding, 'trans' means to change and 'pose' means position. So, transpose of a matrix is obtained by changing the position of its rows and columns.

In other words, rewriting the matrix where the rows become columns and columns become rows is called transpose of matrix.

Let A be a matrix. Transpose of Matrix A is denoted by A' or A^t.

Here is an example:

Transpose of Matrix

Multiplication of Matrices:

Scalar Multiplication:

In scalar multiplication, you are multiplying a matrix by a real number.

When you multiply matrices with real numbers then your result is also a matrix.

Let 'r' be a real number and 'A' be the matrix, the result of scalar multiplication is rA.

Matrix Multiplication:

In Matrix multiplication, you are multiplying one matrix with another matrix. Let A be one matrix and B be another matrix, the product is AB or BA.

Where, AB is not equal to BA.

Remember these things:

To multiply two matrices, the number of columns in the first matrix should be equal to the number of rows in the other matrix.
The product of multiplication of matrices has the rows equal to the number of rows of first matrix and column equal to the number of columns of second matrix.

Have a look at a solved example.

AB = A_2x2 * B_2x1

Multiplication of Matrices

In matrix A, the number of columns is two and in matrix B, the number of rows is 2. They are equal. This satisfies our condition mentioned above and we can multiply the matrices.

Now, the result of the matrix will be AB_2x1. This

also satisfies the above condition.

You saw the example of multiplying two matrices. Some of you might have understood it while some may not.

While multiplying first row and first column of 2x2 matrices:

1. Multiply the first element of first row of Matrix A (a₁₁) with the first element of first column of Matrix B (b₁₁).

2. Multiply the second element of first row of Matrix A (a₁₂) with the second element of first column of Matrix B (b₂₁).

3. Now, add the products of step 1 and step 2.

As a result you will obtain: (a₁₁xb₁₁ + a₁₂xb₂₁).

With this idea, you will have to perform the matrix multiplication as follows:

Multiplication of Matrices based on notation of elements

Now, follow these steps to perform matrix multiplication:

In these steps, A represents matrix A and B represents matrix B.

Multiply first row and first column
Multiply first row and second column
Multiply second row and first column
Multiply second row and second column

Properties of Matrix Multiplication:

Associative Property:

When A,B and C are three matrices where we can perform (AB)C and A(BC), then both results are same i.e. (AB)C = A(BC)

Distributive Property of Matrix Multiplication Over Addition:

When A, B and C are three matrices where we can perform A(B+C) and AB + AC, then both results are same i.e. A(B+C) = AB +AC.

Identity Property:

This property states that matrix A of any order (lxm) is equal to AI, where I is the identity matrix of the same order (lxm). This is also equal to IA.

So, AI = A = IA

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Matrix - Operations on Matrices and their Properties - Unit 3 | Class 09