Question

Find the matrix A satisfying the matrix equation:
$\left[\begin{array}{ccc}2&1\\3&2\\\end{array}\right]$ . A$\left[\begin{array}{ccc}-3&2\\5 &-3\\\end{array}\right]$ = $\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]$

Please help with this question anyone

Answer 1

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

Given matrix equation is

$\rm :\longmapsto\:\rm \: \bigg[ \begin{matrix}2&1 \\ 3&2 \end{matrix} \bigg]A\bigg[ \begin{matrix} - 3&2 \\ 5& - 3 \end{matrix} \bigg] = \bigg[ \begin{matrix}1&0 \\ 0&1 \end{matrix} \bigg]$

Let assume that,

$\rm :\longmapsto\:B = \bigg[ \begin{matrix}2&1 \\ 3&2 \end{matrix} \bigg]$

and

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

So, given matrix equation can be rewritten as

$\rm :\longmapsto\:BAC = I$

Now,

$\red{ \sf{Premultiply \: by \: B^{-1} \: and \: post \: multiply \: by \: {C}^{ - 1}, \: we \: get}}$

So,

$\rm :\longmapsto\: {B}^{ - 1}BA {CC}^{ - 1} = {B}^{ - 1}I {C}^{ - 1}$

$\rm :\longmapsto\:IAI = {B}^{ - 1} {C}^{ - 1}$

$\rm \implies\:\boxed{ \tt{ \: A = {B}^{ - 1} {C}^{ - 1} \: }} - - - (1)$

Now,

We know,

$\sf \: If \: A = \bigg[ \begin{matrix} a&b \\ c&d \end{matrix} \bigg], \: then \: adjA = \bigg[ \begin{matrix} d& - b \\ - c&a \end{matrix} \bigg]$

and

$\rm :\longmapsto\: |A| = ad - bc$

So,

$\rm :\longmapsto\: {B}^{ - 1}$

$\rm = \: \dfrac{1}{ |B| } adjB$

$\rm = \: \dfrac{1}{4 - 3} \bigg[ \begin{matrix}2& - 1 \\ - 3&2 \end{matrix} \bigg]$

$\rm = \: \dfrac{1}{1} \bigg[ \begin{matrix}2& - 1 \\ - 3&2 \end{matrix} \bigg]$

$\rm = \: \bigg[ \begin{matrix}2& - 1 \\ - 3&2 \end{matrix} \bigg]$

$\rm \implies\:\boxed{ \tt{ \: {B}^{ - 1} = \: \bigg[ \begin{matrix}2& - 1 \\ - 3&2 \end{matrix} \bigg] \: }}$

Now,

$\rm :\longmapsto\: {C}^{ - 1}$

$\rm = \: \dfrac{1}{ |C| } adjC$

$\rm = \: \dfrac{1}{9 - 10} \bigg[ \begin{matrix} - 3& - 2 \\ - 5& - 3 \end{matrix} \bigg]$

$\rm = \: \dfrac{1}{ - 1} \bigg[ \begin{matrix} - 3& - 2 \\ - 5& - 3 \end{matrix} \bigg]$

$\rm = \: \bigg[ \begin{matrix} 3& 2 \\ 5& 3 \end{matrix} \bigg]$

$\rm \implies\:\boxed{ \tt{ \: {C}^{ - 1} \: = \: \bigg[ \begin{matrix} 3& 2 \\ 5& 3 \end{matrix} \bigg] \: }}$

Now, On substituting the values in

$\rm \implies\:\boxed{ \tt{ \: A = {B}^{ - 1} {C}^{ - 1} \: }}$

We get

$\rm :\longmapsto\:A = \bigg[ \begin{matrix}2& - 1 \\ - 3&2 \end{matrix} \bigg] \times \bigg[ \begin{matrix} 3& 2 \\ 5& 3 \end{matrix} \bigg]$

$\rm :\longmapsto\:A =\bigg[ \begin{matrix} 6 - 5& 4 - 3 \\ 10 - 9& - 6 + 6 \end{matrix} \bigg]$

$\rm \implies\:\boxed{ \tt{ \: A = \bigg[ \begin{matrix} 1& 1 \\ 1& 0 \end{matrix} \bigg] \: }}$