Find the Adjoint of the Following Matrix: [ − 3 5 2 4 ] Verify that (Adj A) a = |A| I = a (Adj A) for the Above Matrix. - Mathematics

Question

Find the adjoint of the following matrix:
\[\begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]

Verify that (adj A) A = |A| I = A (adj A) for the above matrix.

Solution 1

Given below is the square matrix. Here, we will interchange the diagonal elements and change the signs of the off-diagonal elements.
\[\ A = \begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
\[adjA = \begin{bmatrix}4 & - 5 \\ - 2 & - 3\end{bmatrix}\]
\[(adjA)A = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[\left| A \right| = - 22\]
\[\left| A \right|I = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[A(adjA) = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[ \therefore (adjA)A = \left| A \right|I = A(adjA)\]
Hence verified.

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Solution 2

Given below is the square matrix. Here, we will interchange the diagonal elements and change the signs of the off-diagonal elements.
\[\ A = \begin{bmatrix}- 3 & 5 \\ 2 & 4\end{bmatrix}\]
\[adjA = \begin{bmatrix}4 & - 5 \\ - 2 & - 3\end{bmatrix}\]
\[(adjA)A = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[\left| A \right| = - 22\]
\[\left| A \right|I = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[A(adjA) = \begin{bmatrix}- 22 & 0 \\ 0 & - 22\end{bmatrix}\]
\[ \therefore (adjA)A = \left| A \right|I = A(adjA)\]
Hence verified.

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

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RD Sharma Mathematics [English] Class 12

Chapter 7 Adjoint and Inverse of a Matrix
Exercise 7.1 | Q 1.1 | Page 22

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