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Question

Mathematics Question on Matrices

Find the inverse of each of the matrices, if it exists. [21\11]\begin{bmatrix}2&1\\\1&1\end{bmatrix}

Answer

Let A= [21\11]\begin{bmatrix}2&1\\\1&1\end{bmatrix}
We know that A = IA
so [21\11]\begin{bmatrix}2&1\\\1&1\end{bmatrix}= [10\01]A\begin{bmatrix}1&0\\\0&1\end{bmatrix}A

⇒[10\11]=[1−1\01]A\Rightarrow\begin{bmatrix}1&0\\\1&1\end{bmatrix}=\begin{bmatrix}1&-1\\\0&1\end{bmatrix}A (R1->R1-R2)

⇒[10\01]=[1−1−12]A\Rightarrow\begin{bmatrix}1&0\\\0&1\end{bmatrix}=\begin{bmatrix}1&-1\\\\-1&2\end{bmatrix}A (R2->R2-R1)

∴\therefore A-1= [1−1−12]\begin{bmatrix}1&-1\\\\-1&2\end{bmatrix}