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Find the inverse of the matrix (if it exists) given \(\left[ {\begin{array}{*{20}{c}} { - 1}&5 \\ { - 3}&2 \end{array}} \right]\)

Answer

Let \(A = \left[ {\begin{array}{*{20}{c}} { - 1}&5 \\ { - 3}&2 \end{array}} \right]\)
\(\therefore \left| A \right| = \left| {\begin{array}{*{20}{c}} { - 1}&5 \\ { - 3}&2 \end{array}} \right| \) = -2 - (-15) = -2 + 15 = 13 \( \ne \) 0
\(\therefore\) Matrix A is non-singular and hence A-1 exist.
Now adj. A \(= \left[ {\begin{array}{*{20}{c}} 2&{ - 5} \\ 3&{ - 1} \end{array}} \right]\)And \({A^{ - 1}} = \frac{1}{{\left| A \right|}}adj.A = \frac{1}{{13}}\left[ {\begin{array}{*{20}{c}} 2&{ - 5} \\ 3&{ - 1} \end{array}} \right]\)
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