InterviewSolution
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InterviewSolution
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Find the adjoint of the Matrices.\(\begin{bmatrix}cos\,a&sin\,a\\sin\,a&cos\,a\end{bmatrix}\)Verify that (adj A) A=|A| I=A (adj A) for the above matrices. |
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Answer» A= \(\begin{bmatrix}cos\alpha&sin\alpha\\sin\alpha&cos\alpha\end{bmatrix}\) Cofactors of A are C11 = \(cos\alpha\) C12 = \(-sin\alpha\) C21 = \(-sin\alpha\) C22 =\(cos\alpha\) Since, adj A = \(\begin{bmatrix}C_{11}&C_{12}\\C_{21}&C_{22}\end{bmatrix}^T\) (adj A) =\(\begin{bmatrix}cos\alpha&-sin\alpha\\-sin\alpha&cos\alpha\end{bmatrix}^T\) \(=\begin{bmatrix}cos\alpha&-sin\alpha\\-sin\alpha&cos\alpha\end{bmatrix}\) Now, (adj A)A \(=\begin{bmatrix}cos\alpha&-sin\alpha\\-sin\alpha&cos\alpha\end{bmatrix}\)\(\begin{bmatrix}cos\alpha&sin\alpha\\sin\alpha&cos\alpha\end{bmatrix}\) \(=\begin{bmatrix}-sin^2\alpha+cos^2\alpha&cos\alpha.sin\alpha-sin\alpha.cos\alpha\\-cos\alpha sin\alpha+sin\alpha coa\alpha&-sin^2\alpha+cos^2\alpha\end{bmatrix}\) (adj A)A =\(\begin{bmatrix}cos2\alpha&0\\0&cos2\alpha\end{bmatrix}\) And, |A|.I =\(\begin{bmatrix}cos\alpha&sin\alpha\\sin\alpha&cos\alpha\end{bmatrix}\)\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\) \(=(cos^2\alpha-sin^2\alpha)\)\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\) \(=\begin{bmatrix}cos^2\alpha-sin^2\alpha&0\\0&cos^2\alpha-sin^2\alpha\end{bmatrix}\) \(=\begin{bmatrix}cos2\alpha&0\\0&cos2\alpha\end{bmatrix}\) Also, A(adj A) =\(\begin{bmatrix}cos\alpha&sin\alpha\\sin\alpha&cos\alpha\end{bmatrix}\)\(\begin{bmatrix}cos\alpha&-sin\alpha\\-sin\alpha&cos\alpha\end{bmatrix}\)\(=\begin{bmatrix}cos^2\alpha-sin^2\alpha&0\\0&cos^2\alpha-sin^2\alpha\end{bmatrix}\) \(=\begin{bmatrix}cos2\alpha&0\\0&cos2\alpha\end{bmatrix}\) Hence, (adj A)A = |A|.I = A.(adj A) |
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