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Q.15) Find the inverse of a matrix \( A \) if\[A=\left[\begin{array}{ll}2 & 3 \\5 & 7\end{array}\right]\] |
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Answer» \(A=\begin{bmatrix}2&3\\5&7\end{bmatrix}\) \(|A|=\begin{vmatrix}2&3\\5&7\end{vmatrix}\) = 2 x 7 - 5 x 3 = 14 - 15 = -1 ∵ Aij = (-1)i+j Mij ∴ A11 = (-1)1+1 M11 = 7, A12 = (-1)1+2 M12 = -5, A21 = (-1)2+1 M21 = -3, A22 = (-1)2+2 M22 = 2 Cofactor matrix = CA \(=\begin{bmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{bmatrix}=\begin{bmatrix}7&-5\\-3&2\end{bmatrix}\) Adjoint matrix of A \(=C^T_A=\begin{bmatrix}7&-5\\-3&2\end{bmatrix}^T=\begin{bmatrix}7&-3\\-5&2\end{bmatrix}\) \(A^{-1}=\frac{adj(A)}{|A|}=\frac1{-1}\begin{bmatrix}7&-3\\-5&2\end{bmatrix}=\begin{bmatrix}-7&3\\5&-2\end{bmatrix}\) Hence \(A^{-1}=\begin{bmatrix}-7&3\\5&-2\end{bmatrix}.\) |
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