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Question

Mathematics Question on Matrices

Find the inverse of each of the matrices, if it exists.
[310 27]\begin{bmatrix} 3 & 10\\\ 2 & 7\end{bmatrix}

Answer

Let A=[310 27]\begin{bmatrix} 3 & 10\\\ 2 & 7\end{bmatrix}

We know that A=IAA = IA

[310 27]\begin{bmatrix} 3 & 10\\\ 2 & 7\end{bmatrix}= [10 01]\begin{bmatrix} 1 & 0\\\ 0 & 1\end{bmatrix}A

⇒ [13 27]\begin{bmatrix} 1 & 3\\\ 2 & 7\end{bmatrix}= [1−1 01]\begin{bmatrix} 1 & -1\\\ 0 & 1\end{bmatrix}A (R1→R1−R2)(R_1\rightarrow R_1-R_2)

⇒ [13 01]\begin{bmatrix} 1 & 3\\\ 0 & 1\end{bmatrix}= [1−1 −23]\begin{bmatrix} 1 & -1\\\ -2 & 3\end{bmatrix} A (R2→R2−2R2)(R_2\rightarrow R_2-2R_2)

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

∴A−1=\therefore A^{-1}= [7−10 −23]\begin{bmatrix} 7 & -10\\\ -2 & 3\end{bmatrix}