If a = ⎡ ⎢ ⎣ − 4 − 3 − 3 1 0 1 4 4 3 ⎤ ⎥ ⎦ , Show that Adj a = A. - Mathematics

प्रश्न

If \[A = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix}\], show that adj A = A.

उत्तर

\[A = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix}\]
Now,
\[ C_{11} = \begin{vmatrix}0 & 1 \\ 4 & 3\end{vmatrix} = - 4, C_{12} = - \begin{vmatrix}1 & 1 \\ 4 & 3\end{vmatrix} = 1\text{ and }C_{13} = \begin{vmatrix}1 & 0 \\ 4 & 4\end{vmatrix} = 4\]
\[ C_{21} = - \begin{vmatrix}- 3 & - 3 \\ 4 & 3\end{vmatrix} = - 3, C_{22} = \begin{vmatrix}- 4 & - 3 \\ 4 & 3\end{vmatrix} = 0\text{ and }C_{23} = - \begin{vmatrix}- 4 & - 3 \\ 4 & 4\end{vmatrix} = 4\]
\[ C_{31} = \begin{vmatrix}- 3 & - 3 \\ 0 & 1\end{vmatrix} = - 3, C_{32} = - \begin{vmatrix}- 4 & - 3 \\ 1 & 1\end{vmatrix} = 1\text{ and }C_{33} = \begin{vmatrix}- 4 & - 3 \\ 1 & 0\end{vmatrix} = 3\]
\[ \therefore adj A = \begin{bmatrix}- 4 & 1 & 4 \\ - 3 & 0 & 4 \\ - 3 & 1 & 3\end{bmatrix}^T = \begin{bmatrix}- 4 & - 3 & - 3 \\ 1 & 0 & 1 \\ 4 & 4 & 3\end{bmatrix} = A\]
Hence proved .

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Properties of Matrix Multiplication - Inverse of a Square Matrix by the Adjoint Method

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पाठ 7: Adjoint and Inverse of a Matrix - Exercise 7.1 [पृष्ठ २२]

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आरडी शर्मा Mathematics [English] Class 12

पाठ 7 Adjoint and Inverse of a Matrix
Exercise 7.1 | Q 4 | पृष्ठ २२

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