1.

Find the adjoint of the Matrices.\(\begin{bmatrix}-3&5\\2&4\end{bmatrix}\)Verify that (adj A) A=|A| I=A (adj A) for the above matrices.

Answer»

A = \(\begin{bmatrix}-3&5\\2&4\end{bmatrix}\)

Cofactors of A are 

C11 = 4 

C12 = – 2 

C21 = – 5 

C22 = – 3 

Since, adj A = \(\begin{bmatrix}C_{11}&C_{12}\\2_{21}&C_{22}\end{bmatrix}^T\)

(adj A) = \(\begin{bmatrix}4&-2\\-5&-3\end{bmatrix}^T\)

\(=\begin{bmatrix}4&-5\\-2&-3\end{bmatrix}^T\)

Now, (adj A)A \(=\begin{bmatrix}4&-5\\-2&-3\end{bmatrix}\)\(\begin{bmatrix}-3&5\\2&4\end{bmatrix}\)\(=\begin{bmatrix}-12&-10&20&-20\\6&-6&-10&-12\end{bmatrix}\)

(adj A)A \(=\begin{bmatrix}-22&0\\0&-22\end{bmatrix}\)

And, |A|.I =\(\begin{bmatrix}-3&5\\2&4\end{bmatrix}\)\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)=(-22)\(\begin{bmatrix}1&0\\0&1\end{bmatrix}\)\(=\begin{bmatrix}-22&0\\0&-22\end{bmatrix}\)

Also, A(adj A) = \(\begin{bmatrix}-3&5\\2&4\end{bmatrix}\)\(\begin{bmatrix}4&-5\\-2&-3\end{bmatrix}\)\(=\begin{bmatrix}-12&-10&20&-20\\6&-6&-10&-12\end{bmatrix}\)

A(adj A) \(=\begin{bmatrix}-22&0\\0&-22\end{bmatrix}\)

Hence, (adj A)A = |A|.I = A.(adj A)


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