SCALAR MULTIPLICATION OF MATRICES

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Scalar multiplication of matrices is the process of multiplying a matrix by a constant or real number.

It is the process of multiplying a real number to a matrix to get a scalar multiple.

A Scalar multiple is an entry of a matrix that results from scalar multiplication.

The process of scalar multiplication is exemplified as:

If $A=\left[{\begin{array}{cc} a & b \\ c & d  \\ \end {array} } \right]$, then the scalar multiple 3A is

$$3A=\left[{\begin{array}{cc} 3×a & 3×b \\ 3×c & 3×d  \\ \end {array} } \right]$$

$$3A=\left[{\begin{array}{cc} 3a & 3b \\ 3c & 3d  \\ \end {array} } \right]$$

Scalar multiplication simply means you should multiply each entry in the matrix by the constant.

Example 1

Multiply Matrix A by the scalar $-5$ given that:

$A=\left[{\begin{array}{cc} 4 & 5 \\ 3 & 9 \\ \end {array} } \right]$

Solution:

$-5A=\left[{\begin{array}{cc} -5×4 & -5×5 \\ -5×3 & -5×9  \\ \end {array} } \right]$

$-5A=\left[{\begin{array}{cc} -20 & -25 \\ -15 & -45  \\ \end {array} } \right]$

Example 2

Find $-4A+3B$ given that

$A=\left[{\begin{array}{cc} 4 & 5 \\ 3 & 9 \\ \end {array} } \right]$ and $B=\left[{\begin{array}{cc} 6 & 7 \\ 3 & 2 \\ \end {array} } \right]$ 

Solution:

First, let's determine $-4A$

$-4A=\left[{\begin{array}{cc} -4×4 & -4×5 \\ -4×3 & -4×9 \\ \end {array} } \right]$ 

$-4A=\left[{\begin{array}{cc} -16 & -20 \\ -12 & -36 \\ \end {array} } \right]$ 

Now, let's determine 3B

$3B=\left[{\begin{array}{cc} 3×6 & 3 ×7 \\ 3×3 & 3×2 \\ \end {array} } \right]$ 

$3B=\left[{\begin{array}{cc} 18 & 21 \\ 9 & 6 \\ \end {array} } \right]$ 

Now, let's determine $-4A+3B$

$-4A+3B=\left[{\begin{array}{cc} -16+18 & -20 +21 \\ -12+9 & -36+6 \\ \end {array} } \right]$ 

$-4A+3B=\left[{\begin{array}{cc} 2 & 1 \\  -3 & -30 \\ \end {array} } \right]$

Example 3

Find the scalar multiple of 9A given that

$A=\left[{\begin{array}{cccc} 4 & 5 & 3 \\ 3 & 9 & 2 \\ 5 & -5 & 0\\ \end {array} } \right]$

Solution:

$9A=\left[{\begin{array}{cccc} 9×4 & 9×5 & 9×3 \\ 9×3 & 9×9 & 9×2 \\ 9×5 & 9×(-5) & 9×0\\ \end {array} } \right]$

$9A=\left[{\begin{array}{cccc} 36 & 45 & 27 \\ 27 & 81 & 18 \\ 45 & -45 & 0\\ \end {array} } \right]$

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