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Question

Mathematics Question on Matrices

Find the inverse of each of the matrices, if it exists. [13 27]\begin{bmatrix} 1 & 3\\\ 2 & 7\end{bmatrix}

Answer

Let A=$$\begin{bmatrix} 1 & 3\\\ 2 & 7\end{bmatrix}

We know that A=IAA = IA

so[13 27]\begin{bmatrix} 1 & 3\\\ 2 & 7\end{bmatrix}= [10 01]A\begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix}A

⇒ [13 01]\begin{bmatrix} 1 & 3\\\ 0 & 1\end{bmatrix}= [10 −21]A\begin{bmatrix} 1 & 0\\\ -2 & 1\end{bmatrix}A (R2→R2−2R1)(R_2\rightarrow R_2-2R_1)

⇒ [10 01]=[7−3 −21]A\begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix}=\begin{bmatrix} 7 & -3\\\ -2 & 1 \end{bmatrix}A (R1→R1−3R2)(R_1\rightarrow R_1-3R_2)

∴A−1\therefore A^{-1} =[7−3 −21]\begin{bmatrix} 7 & -3\\\ -2 & 1 \end{bmatrix}