# Scalar Multiplication with Matrices – Properties and Examples

Here you will learn scalar multiplication with matrices (multiplicaition of a matrix by a scalar ) and properties of scalar mutiplication.

Let’s begin –

## Scalar Multiplication with Matrices

Definition : Let $$[a_{ij}]$$ be an $$m\times n$$ matrix and k be any number called a scalar. Then the matrix obtained by mutiplying every element of A by k is called the scalar multiple of A by k and is denoted by kA.

Thus,

kA = $$[ka_{ij}]_{m\times n}$$

Example : if A = $$\begin{bmatrix} 1 & 2 & 5 \\ -2 & 3 & 4 \\ 1 & 2 & -1 \end{bmatrix}$$, then 3A = $$\begin{bmatrix} 3 & 6 & 15 \\ -6 & 9 & 12 \\ 3 & 6 & -3 \end{bmatrix}$$

if A = $$\begin{bmatrix} 6 & 2 & 3 \\ 2 & 3 & -2 \\ 2 & 4 & 1 \end{bmatrix}$$, then $$1\over 2$$A = $$\begin{bmatrix} 3 & 1 & 3/2 \\ 1 & 3/2 & -1 \\ 1 & 2 & 1/2 \end{bmatrix}$$

Example :  Let A = $$\begin{bmatrix} 1 & 5 & 7 & 3\\ -1 & 5 & 9 & 4 \\ -2 & 6 & 3 & -5 \end{bmatrix}$$, then 2A = $$\begin{bmatrix} 2 & 10 & 14 & 6\\ -2 & 10 & 18 & 8 \\ -4 & 12 & 6 & -10 \end{bmatrix}$$

and $$1\over 2$$A = $$\begin{bmatrix} 1/2 & 5/2 & 7/2 & 3/2 \\ -1/2 & 5/2 & 9/2 & 2 \\ -1 & 3 & 3/2 & -5/2 \end{bmatrix}$$

#### Properties of scalar multiplication

Various properties of scalar multiplication are stated below :

If A = $$[a_{ij}]_{m\times n}$$, B = $$[b_{ij}]_{m\times n}$$ are two matrices and k and l are scalars, then

(i) k (A + B) = k A + k B

(ii) (k + l) A = k A + l A

(iii)  (k l) A = k (l A) = l (k A)

(iv) (-k) A = –  (k A) = k(-A)

(v) 1 A = A

(vi) (-1) A = -A