If $\displaystyle P\ =\ \begin{bmatrix}2 & 4\\ 3 & 5\end{bmatrix} \ and\ Q\ =\ \begin{bmatrix}-2 & 2\\ 4 & 1\end{bmatrix}$ , find the marix R such hat $P - Q + R$ is an Identity matrix.

Given :

$P=\begin{bmatrix} 2 & 4\\ 3 & 5 \end{bmatrix} \ and\ Q=\begin{bmatrix} -2 & 2\\ 4 & 1 \end{bmatrix}$

$P - Q + R$

To find :

We have to find the matrix R.

Solution :

Let $ R=\begin{bmatrix} a & b\\ c & d \end{bmatrix}$

Identity matrix $ I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

$P-Q+R=I$ 

LHS

$P-Q+R=\begin{bmatrix} 2 & 4\\ 3 & 5 \end{bmatrix} -\begin{bmatrix} -2 & 2\\ 4 & 1 \end{bmatrix} +\begin{bmatrix} a & b\\ c & d \end{bmatrix}$

$=\begin{bmatrix} 2-( -2) +a & 4-2+b\\ 3-4+c & 5-1+d \end{bmatrix}$

$ =\begin{bmatrix} 2+2+a & 2+b\\ -1+c & 4+d \end{bmatrix}$

$ =\begin{bmatrix} 4+a & 2+b\\ c-1 & 4+d \end{bmatrix}$

RHS

$I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

$\begin{bmatrix} 4+a & 2+b\\ c-1 & 4+d \end{bmatrix} =\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

This implies,

$4+a=1$

$a=1-4=-3$

$2+b=0$

$b=-2$

$c-1=0$

$c=1$

$4+d=1$

$d=1-4=-3$

Therefore,

$R=\begin{bmatrix} -3 & -2\\ 1 & -3 \end{bmatrix}$.

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Updated on: 10-Oct-2022

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