In this session, we will explore the fundamental operations performed on matrices and observe the algebraic properties that govern matrix addition and subtraction using numerical examples.

a) Multiplication by Scalar

When a matrix is multiplied by a scalar (a real number), every individual element inside the matrix is multiplied by that number.

b) Addition and Subtraction of Matrices

Two matrices can be added or subtracted if and only if they have the same order. The operation is performed by adding or subtracting the corresponding elements.

Let us define three $2 \times 2$ matrices to observe the following properties:

$$A = \begin{pmatrix} 2 & 5 \\ 1 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & -1 \\ 0 & 2 \end{pmatrix}, \quad C = \begin{pmatrix} -2 & 4 \\ 5 & 1 \end{pmatrix}$$

1. Closure Property (Addition and Subtraction)

Proof for Addition:

Proof for Subtraction:

2. Commutative Property (Addition)

Proof:

First, calculate $\text{L.H.S.} (A + B)$:

Next, calculate $\text{R.H.S.} (B + A)$:

3. Associative Property (Addition)

Proof:

Calculate $\text{L.H.S. } ((A + B) + C)$:

We already know $A + B = \begin{pmatrix} 5 & 4 \\ 1 & 6 \end{pmatrix}$. Now add $C$:

Calculate $\text{R.H.S. } (A + (B + C))$:

First find $(B + C)$:

Now add $A$ to $(B + C)$:

4. Additive Inverse

Proof:

Given $A = \begin{pmatrix} 2 & 5 \\ 1 & 4 \end{pmatrix}$, its negative matrix is $-A = \begin{pmatrix} -2 & -5 \\ -1 & -4 \end{pmatrix}$.

5. Additive Identity

Proof:

Let $\mathbf{O} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ and $A = \begin{pmatrix} 2 & 5 \\ 1 & 4 \end{pmatrix}$.