Scalar Multiplication of a Matrix

Just like vectors, matrices can also be multiplied by scalars. It follows the same idea as scalar multiplication of vectors.

Scalar multiplication of a matrix (denoted by $cA$ where $c$ is a scalar and $A$ is a matrix) is defined as follows:

For each row $i$ and each column $j$ of some matrix $A$ of size $m \times n$ ,

[cA]_{ij} = c[A]_{ij}, \quad 1 \le i \le m, \; 1 \le j \le n.

What this means is that to compute the element $ij$ of the matrix $cA$ , you simply multiply the element $ij$ of $A$ by $c$ .

Another way of writing this would be

cA = \begin{bmatrix} c a_{11} & c a_{12} & \cdots & c a_{1n} \\ c a_{21} & c a_{22} & \cdots & c a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ c a_{m1} & c a_{m2} & \cdots & c a_{mn} \end{bmatrix}.

This is similar to what you do when multiplying a vector by a scalar.

In conclusion, scalar multiplication of a matrix scales every entry of the matrix by the same factor.

Example

Take the matrix

A = \begin{bmatrix} 2 & -1 & 3 & 0 \\ -4 & 1 & 2 & -2 \\ 5 & -3 & 0 & 1 \end{bmatrix},

and the scalar

c = -2.

We compute $cA$ :

\begin{align*} cA &= -2 \begin{bmatrix} 2 & -1 & 3 & 0 \\ -4 & 1 & 2 & -2 \\ 5 & -3 & 0 & 1 \end{bmatrix} \\ &= \begin{bmatrix} (-2) \cdot 2 & (-2) \cdot -1 & (-2) \cdot 3 & (-2) \cdot 0 \\ (-2) \cdot (-4) & (-2) \cdot 1 & (-2) \cdot 2 & (-2) \cdot (-2) \\ (-2) \cdot 5 & (-2) \cdot (-3) & (-2) \cdot 0 & (-2) \cdot 1 \end{bmatrix} \\ &= \begin{bmatrix} -4 & 2 & -6 & 0 \\ 8 & -2 & -4 & 4 \\ -10 & 6 & 0 & -2 \end{bmatrix} \\ \end{align*}

Just like addition and subtraction of matrices, scalar multiplication also inherits the algebraic properties of regular scalar multiplication.

For a matrix $A$ of size $m \times n$ and for scalars $c$ , $cA$ is also a matrix of size $m \times n$ .

For some matrix $A$ and for scalars $c$ and $d$ ,

(cd)A = c(dA).

For any matrix $A$ , $1A = A$ .

For two $m \times n$ matrices $A$ and $B$ and a scalar $c$ ,

c(A + B) = cA + cB.

Also,

(c + d)A = cA + dA.