Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 2 & -3\\ -1 & 2 \end{bmatrix}\)
Solution and Explanation
Let A=\(\begin{bmatrix} 2 & -3\\ -1 & 2 \end{bmatrix}\)
We know that A = IA
\(\begin{bmatrix} 2 & -3\\ -1 & 2 \end{bmatrix}\)=A\(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\)
⇒ \(\begin{bmatrix} 1 & -1\\ -1 & 2 \end{bmatrix}\)= \(\begin{bmatrix} 1 & 1\\ 0 & 1 \end{bmatrix}\)A \((R_1\rightarrow R_1+R_2)\)
⇒ \(\begin{bmatrix} 1 & -1\\ 0 & 1 \end{bmatrix}\)=\(\begin{bmatrix} 1 & 1\\ 1 & 2 \end{bmatrix}\)A \((R_2\rightarrow R_2+R_1)\)
⇒ \(\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}\)= \(\begin{bmatrix} 2 & 3\\ 1 & 2 \end{bmatrix}\)A \((R_1\rightarrow R_1+R_2)\)
so A-1=\(\begin{bmatrix} 2 & 3\\ 1 & 2 \end{bmatrix}\)
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