Question

Find X and Y if X + Y =
$\binom{7 \: \: 0}{2 \: \: 5}$
and X - Y =
$\binom{3 \: \: 0}{0 \: \: 3}$
​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that

$\rm :\longmapsto\:X + Y = \bigg[ \begin{matrix}7&0 \\ 2&5 \end{matrix} \bigg] - - - (1)$

and

$\rm :\longmapsto\:X - Y = \bigg[ \begin{matrix}3&0 \\ 0&3 \end{matrix} \bigg] - - - (2)$

On Adding equation (1) and equation (2), we get

$\rm :\longmapsto\:X + Y + X - Y =\bigg[ \begin{matrix}7&0 \\ 2&5 \end{matrix} \bigg] + \bigg[ \begin{matrix}3&0 \\ 0&3 \end{matrix} \bigg]$

$\rm :\longmapsto\:2X = \bigg[ \begin{matrix}7 + 3&0 + 0 \\ 2 + 0&5 + 3 \end{matrix} \bigg]$

$\rm :\longmapsto\:2X = \bigg[ \begin{matrix}10&0 \\ 2&8 \end{matrix} \bigg]$

$\bf :\longmapsto\:X = \bigg[ \begin{matrix}5&0 \\ 1&4 \end{matrix} \bigg]$

On Subtracting equation (2) from equation (1), we get

$\rm :\longmapsto\:X + Y - X + Y =\bigg[ \begin{matrix}7&0 \\ 2&5 \end{matrix} \bigg] - \bigg[ \begin{matrix}3&0 \\ 0&3 \end{matrix} \bigg]$

$\rm :\longmapsto\:2Y = \bigg[ \begin{matrix}7 - 3&0 - 0 \\ 2 - 0&5 - 3 \end{matrix} \bigg]$

$\rm :\longmapsto\:2Y = \bigg[ \begin{matrix}4&0 \\ 2&2 \end{matrix} \bigg]$

$\bf :\longmapsto\:Y = \bigg[ \begin{matrix}2&0 \\ 1&1 \end{matrix} \bigg]$

Hence,

$\bf :\longmapsto\:X = \bigg[ \begin{matrix}5&0 \\ 1&4 \end{matrix} \bigg]$

and

$\bf :\longmapsto\:Y = \bigg[ \begin{matrix}2&0 \\ 1&1 \end{matrix} \bigg]$

Additional Information :-

1. Order of a matrix is defined as number of rows × number of columns. Order represents the following information :

(a) Number of elements in matrix

(b) Number of elements in each row (= number of columns)

(c) Number of elements in each column (= number of rows)

2. Addition of matrices is possible only when order of matrices are same otherwise matrix addition is meaningless.

3. Matrix multiplication is defined when number of columns of pre - multiplier is equal to number of rows of post - multiplier.