Math Doubts

Scalar multiplication Rule of matrices

Formula

$k \times \big[\,e_{ij}\,\big] \,=\, \big[k \times e_{ij}\big]$

The multiplication of a matrix by a scalar is equal to the multiplication of each entry (or element) by the scalar in the matrix, is called the scalar multiplication rule of the matrices.

Introduction

scalar multiplication matrix

Let $k$ be a constant and the matrix $M$ represents a matrix of the order $m \times n$.

$M$ $\,=\,$ $\begin{bmatrix} e_{11} & e_{12} & e_{13} & \cdots & e_{1n}\\ e_{21} & e_{22} & e_{23} & \cdots & e_{2n}\\ e_{31} & e_{32} & e_{33} & \cdots & e_{3n}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ e_{m1} & e_{m2} & e_{m3} & \cdots & e_{mn} \end{bmatrix}$

Let’s assume that the matrix $M$ is multiplied by the scalar $k$.

$\implies$ $k \times M$ $\,=\,$ $k \times \begin{bmatrix} e_{11} & e_{12} & e_{13} & \cdots & e_{1n}\\ e_{21} & e_{22} & e_{23} & \cdots & e_{2n}\\ e_{31} & e_{32} & e_{33} & \cdots & e_{3n}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ e_{m1} & e_{m2} & e_{m3} & \cdots & e_{mn} \end{bmatrix}$

The product of a scalar and a matrix is equal to the product of each element and scalar in a matrix.

$\implies$ $k \times M$ $\,=\,$ $\begin{bmatrix} k \times e_{11} & k \times e_{12} & k \times e_{13} & \cdots & k \times e_{1n}\\ k \times e_{21} & k \times e_{22} & k \times e_{23} & \cdots & k \times e_{2n}\\ k \times e_{31} & k \times e_{32} & k \times e_{33} & \cdots & k \times e_{3n}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ k \times e_{m1} & k \times e_{m2} & k \times e_{m3} & \cdots & k \times e_{mn} \end{bmatrix}$

This mathematical property of matrices is called the scalar multiplication of the matrices. It is used as a formula to multiply any matrix by any scalar.

If the matrix $M$ is simply expressed as $\big[e_{ij}\big]$, then the scalar multiplication rule can be simply written as follows.

$\implies$ $k \big[e_{ij}\big] \,=\, \big[k e_{ij}\big]$

Here, the letters $i$ and $j$ represent “the number of the row” and “the number of the column”.

Examples

Let’s learn how to use the scalar multiplication rule of the matrices from some understandable examples.

$(1).\,\,\,$ $4$ $\times$ $\begin{bmatrix} 1 & 7 \\ -2 & 6 \end{bmatrix}$

In this example, the matrix of the order $2$ is multiplied by a scalar $4$. It can be evaluated by multiplying each entry in the matrix by the scalar $4$.

$\implies$ $4$ $\times$ $\begin{bmatrix} 1 & 7 \\ -2 & 6 \end{bmatrix}$ $\,=\,$ $\begin{bmatrix} 4 \times 1 & 4 \times 7 \\ 4 \times (-2) & 4 \times 6 \end{bmatrix}$

$\,\,\,\therefore\,\,\,\,\,\,$ $4$ $\times$ $\begin{bmatrix} 1 & 7 \\ -2 & 6 \end{bmatrix}$ $\,=\,$ $\begin{bmatrix} 4 & 28 \\ -8 & 24 \end{bmatrix}$

$(2).\,\,\,$ $-1$ $\times$ $\begin{bmatrix} 2 & 0 & 3 & 5 \\ 5 & 1 & 8 & 4 \\ 7 & 3 & 2 & 6 \end{bmatrix}$

In this case, a matrix of the order $3 \times 4$ is multiplied by the scalar $-1$. So, multiply each entry in this matrix by the number -1.

$\implies$ $-1$ $\times$ $\begin{bmatrix} 2 & 0 & 3 & 5 \\ 5 & 1 & 8 & 4 \\ 7 & 3 & 2 & 6 \end{bmatrix}$ $\,=\,$ $\begin{bmatrix} (-1) \times 2 & (-1) \times 0 & (-1) \times 3 & (-1) \times 5 \\ (-1) \times 5 & (-1) \times 1 & (-1) \times 8 & (-1) \times 4 \\ (-1) \times 7 & (-1) \times 3 & (-1) \times 2 & (-1) \times 6 \end{bmatrix}$

$\,\,\,\therefore\,\,\,\,\,\,$ $-1$ $\times$ $\begin{bmatrix} 2 & 0 & 3 & 5 \\ 5 & 1 & 8 & 4 \\ 7 & 3 & 2 & 6 \end{bmatrix}$ $\,=\,$ $\begin{bmatrix} -2 & 0 & -3 & -5 \\ -5 & -1 & -8 & -4 \\ -7 & -3 & -2 & -6 \end{bmatrix}$

$(3).\,\,\,$ $6$ $\times$ $\begin{bmatrix} -9 & 5 & -2 \\ 5 & 8 & -3 \\ 6 & -1 & 0 \end{bmatrix}$

Use the scalar multiplication of matrices formula for multiplying the square matrix of the order $3$ by the scalar $6$.

$\implies$ $6$ $\times$ $\begin{bmatrix} -9 & 5 & -2 \\ 5 & 8 & -3 \\ 6 & -1 & 0 \end{bmatrix}$ $\,=\,$ $\begin{bmatrix} 6 \times (-9) & 6 \times 5 & 6 \times (-2) \\ 6 \times 5 & 6 \times 8 & 6 \times (-3) \\ 6 \times 6 & 6 \times (-1) & 6 \times 0 \end{bmatrix}$

$\,\,\,\therefore\,\,\,\,\,\,$ $6$ $\times$ $\begin{bmatrix} -9 & 5 & -2 \\ 5 & 8 & -3 \\ 6 & -1 & 0 \end{bmatrix}$ $\,=\,$ $\begin{bmatrix} -54 & 30 & -12 \\ 30 & 48 & -18 \\ 36 & -6 & 0 \end{bmatrix}$

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