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Question

Mathematics Question on Matrices

Find the inverse of each of the matrices, if it exists. [1−1\23]\begin{bmatrix}1&-1\\\2&3\end{bmatrix}

Answer

Let A= [1−1\23]\begin{bmatrix}1&-1\\\2&3\end{bmatrix} We know that A = IA
∴[1−1\23]=[10\01]A\therefore \begin{bmatrix}1&-1\\\2&3\end{bmatrix}=\begin{bmatrix}1&0\\\0&1\end{bmatrix}A
⇒[1−1\05]=[10−21]A\Rightarrow\begin{bmatrix}1&-1\\\0&5\end{bmatrix}=\begin{bmatrix}1&0\\\\-2&1\end{bmatrix}A (R2 →\to R2-2R1)
⇒[1−1\05]=[10−2515]A\Rightarrow\begin{bmatrix}1&-1\\\0&5\end{bmatrix}=\begin{bmatrix}1&0\\\\-\frac{2}{5}&\frac{1}{5}\end{bmatrix}A (R2 →15R2\to \frac{1}{5R_2})
⇒[1−1\05]=[3515−2515]A\Rightarrow\begin{bmatrix}1&-1\\\0&5\end{bmatrix}=\begin{bmatrix}\frac{3}{5}&\frac{1}{5}\\\\-\frac{2}{5}&\frac{1}{5}\end{bmatrix}A (R1 →\to R1+R2)

so A-1= [3515−2515]\begin{bmatrix}\frac{3}{5}&\frac{1}{5}\\\\-\frac{2}{5}&\frac{1}{5}\end{bmatrix}