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Mathematics Question on Invertible Matrices

The inverse of the matrix [5−2 31 ]\begin{bmatrix} {5}&{-2}\\\ {3}&{1}\\\ \end{bmatrix} is is

A

111[12 −35 ]\frac {1}{11}\begin{bmatrix} {1}&{2}\\\ {-3}&{5}\\\ \end{bmatrix}

B

[12 −35 ]\begin{bmatrix} {1}&{2}\\\ {-3}&{5}\\\ \end{bmatrix}

C

113[−25 13 ]\frac {1}{13}\begin{bmatrix} {-2}&{5}\\\ {1}&{3}\\\ \end{bmatrix}

D

[13 −25 ]\begin{bmatrix} {1}&{3}\\\ {-2}&{5}\\\ \end{bmatrix}

Answer

111[12 −35 ]\frac {1}{11}\begin{bmatrix} {1}&{2}\\\ {-3}&{5}\\\ \end{bmatrix}

Explanation

Solution

Let A=[5−2 31]A=\begin{bmatrix}5 & -2 \\\ 3 & 1\end{bmatrix} ∣A∣=5+6=11|A|=5+6=11 and adj A=[12 −35]adj\, A=\begin{bmatrix}1 & 2 \\\ -3 & 5\end{bmatrix} A−1=1∣A∣A^{-1}=\frac{1}{|A|} (adj A) =111[12 −35]=\frac{1}{11}\begin{bmatrix}1 & 2 \\\ -3 & 5\end{bmatrix}