# Multiplication of a Matrix by a Number

We will discuss here about the process of Multiplication of a matrix by a number.

The multiplication of a matrix A by a number k gives a matrix of the same order as A, in which all the elements are k times the elements of A.

Example:

Let A = $$\begin{bmatrix} 10 & 5\\ -3 & -7 \end{bmatrix}$$ and B = $$\begin{bmatrix} -2 & 9\\ 0 & 3\\ -1 & 5 \end{bmatrix}$$

Then, kA = k$$\begin{bmatrix} 10 & 5\\ -3 & -7 \end{bmatrix}$$

= $$\begin{bmatrix} 10k & 5k\\ -3k & -7k \end{bmatrix}$$ and

kB = k$$\begin{bmatrix} -2 & 9\\ 0 & 3\\ -1 & 5 \end{bmatrix}$$

= $$\begin{bmatrix} -2k & 9k\\ 0 & 3k\\ -1k & 5k \end{bmatrix}$$

Similarly,

$$\begin{bmatrix} a & b\\ c & d \end{bmatrix}$$ = $$\frac{1}{k}$$$$\begin{bmatrix} ka & kb\\ kc & kd \end{bmatrix}$$.

Solved examples on Multiplication of a Matrix by a Number (Scalar Multiplication):

1. If A = $$\begin{bmatrix} 10 & -9\\ -1 & 4 \end{bmatrix}$$, find 4A.

Solution:

4A = 4$$\begin{bmatrix} 10 & -9\\ -1 & 4 \end{bmatrix}$$

= $$\begin{bmatrix} 4 × 10 & 4 × (-9)\\ 4 × (-1) & 4 × 4 \end{bmatrix}$$

= $$\begin{bmatrix} 40 & -36\\ -4 & 16 \end{bmatrix}$$

2. If M = $$\begin{bmatrix} 2 & -3\\ -4 & 5 \end{bmatrix}$$, find -5A.

Solution:

-5M = -5$$\begin{bmatrix} 2 & -3\\ -4 & 5 \end{bmatrix}$$

= $$\begin{bmatrix} (-5) × 2 & (-5) × (-3)\\ (-5) × (-4) & (-5) × 5 \end{bmatrix}$$

= $$\begin{bmatrix} -10 & 15\\ 20 & -25 \end{bmatrix}$$