1.

Find the adjoint of the Matrices.\(\begin{bmatrix}a&b\\c&d\end{bmatrix}\)Verify that (adj A) A=|A| I=A (adj A) for the above matrices.

Answer»

A = \(\begin{bmatrix} a&b \\ c&d \end{bmatrix}\)

Cofactors of A are

C11 = d

C12 = – c

C21 = – b

C22 = a

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

(adj A) = \(\begin{bmatrix}d&-c\\-b&a\end{bmatrix}^T\)

\(=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\)

Now, (adj A)A \(=\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\)\(\begin{bmatrix} a&b \\ c&d \end{bmatrix}\)\(=\begin{bmatrix}ad&-bc&bd&-bd\\-ac&+ac&-bc&+ad\end{bmatrix}\)

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

And, |A|.I =\(\begin{bmatrix} a&b \\ c&d \end{bmatrix}\)\(\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}\) = (ad - bc)\(\begin{bmatrix} 1&0 \\ 0&1 \end{bmatrix}\)= \(\begin{bmatrix}ad&-bc&0\\0&ad&-bc\end{bmatrix}\) 

Also, A(adj A) = \(\begin{bmatrix} a&b \\ c&d \end{bmatrix}\)\(\begin{bmatrix}d&-b\\-c&a\end{bmatrix}\)\(=\begin{bmatrix}ad&-bc&bd&-bd\\-ac&+ac&-bc&+ad\end{bmatrix}\)

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


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