if A= [ ll 1 & 2 3 & 4 ] and X is a 2 2 matrix such that A X=1, then find X.

We will reduce the matrix A to the identity matrix by using row transformations. During this proccess, I will be converted to the matrix X. We have AX = I. [ ll 1 & 2 3 & 4 ] X= [ ll 1 & 0 0 & 1 ]By R_2-3 R_1, we get, [ rr 1 & 2 0 & -2 ] X= [ rr 1 & 0 -3 & 1 ] By (-1 2 ) R_2, we get, [ ll 1 & 2 0 & 1 ] X= [ rr 1 & 0 3 2 & -1 2 ] By R_1-2 R_2, we get, Maharash l [ ll 1 & 0 0 & 1 ]X= [ rr -2 & 1 3 2 & -1 2 ] X= [ rr -2 & 1 3 2 & -1 2 ]

if A= [ ll 1 & 2 3 & 4 ] and X is a 2 2 matrix such that A X=1, t…

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Question

if $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]$ and $X$ is a $2 \times 2$ matrix such that $A X=1$, then find $X$.

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Answer

We will reduce the matrix $A$ to the identity matrix by using row transformations. During this proccess, I will be converted to the matrix $X.$
We have $AX = I.$
$\therefore\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right] X=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$By $R_2-3 R_1$, we get,
$\left[\begin{array}{rr}1 & 2 \\0 & -2\end{array}\right] X=\left[\begin{array}{rr}1 & 0 \\-3 & 1\end{array}\right]$
By $\left(-\frac{1}{2}\right) R_2$, we get,
$\left[\begin{array}{ll}1 & 2 \\0 & 1\end{array}\right] X=\left[\begin{array}{rr}1 & 0 \\\frac{3}{2} & -\frac{1}{2}\end{array}\right]$
By $R_1-2 R_2$, we get, Maharash
$\begin{array}{l}{\left[\begin{array}{ll}1 & 0 \\0 & 1\end{array}\right]X=\left[\begin{array}{rr}-2 & 1 \\\frac{3}{2} & -\frac{1}{2}\end{array}\right]} \\ \therefore X=\left[\begin{array}{rr}-2 & 1 \\\frac{3}{2} & -\frac{1}{2}\end{array}\right]\end{array}$

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