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Find inverse of the following matrices using element (a) \( \left[\begin{array}{rr}7 & 1 \\ 4 & -3\end{array}\right] \) (b) \( \left[\begin{array}{rr}1 & 6 \\ -3 & 5\end{array}\right] \) |
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Answer» (a) \(A = \begin{bmatrix} 7&1\\4&-3 \end{bmatrix}\) \(|A| = \begin{vmatrix} 7&1\\4&-3\end{vmatrix} = -21 -4 = -25\) \(A_{11} = (-1)^{1 + 1} (-3) = -3\) \(A_{12} = (-1)^{1 +2}4 = -4\) \(A_{21} = (-1)^{2 + 1}1 = -1\) \(A_{22} = (-1)^{2 + 2} \,7 = 7\) \(\therefore adj \,A = {C_{ij}}^T = \begin{bmatrix}-3 &-4\\-1&7\end{bmatrix}^T = \begin{bmatrix}-3 &-1\\-4&7\end{bmatrix}\) \(\therefore A^{-1} = \frac{adj\,A}{|A|} = \frac{-1}{25} \begin{bmatrix}-3 &-1\\-4&7\end{bmatrix}\). (b) \(B = \begin{bmatrix}1 &6\\-3&5\end{bmatrix}\) \(|B| = \begin{vmatrix} 1&6\\-3&5\end{vmatrix} = 5+18 = 23\) \(\therefore B^{-1} = \frac{adj\,B}{|B|} = \frac{1}{23} \begin{bmatrix}5 &-6\\3&1\end{bmatrix}\) |
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