Find the inverse of the matrix (if it exists) given [ * 20 c - 1 … - Vidyadip

Question

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

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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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