1.

Find the adjoint of the Matrices and Verify that (adj A) A = |A| I = A (adj A)\(\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}\)Verify that (adj A) A=|A| I=A (adj A) for the above matrices.

Answer»

A = \(\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}\)

Cofactors of A are: 

C11 = – 3 C21 = 2 C31 = 2 

C12 = 2 C22 = – 3 C23 = 2 

C13 = 2 C23 = 2 C33 = – 3

adj A =\(\begin{bmatrix}C_{11}&C_{12}&C_{13}\\C_{21}&C_{22}&C_{23}\\C_{31}&C_{32}&C_{33}\end{bmatrix}^T\)

\(\begin{bmatrix}-3&2&2\\2&-3&2\\2&2&-3\end{bmatrix}\)

Now, (adj A).A =\(\begin{bmatrix}-3&2&2\\2&-3&2\\2&2&-3\end{bmatrix}\)\(\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}\)

\(=\begin{bmatrix}-3+4+4&-6+2+4&-6+4+2\\2-3+4&4-3+4&4-6+2\\2+4-6&4+2-6&4+4-3\end{bmatrix}\)

\(=\begin{bmatrix}5&0&5\\0&5&0\\0&0&5\end{bmatrix}\)

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

Then, A.(adj A) =\(\begin{bmatrix}1&2&2\\2&1&2\\2&2&1\end{bmatrix}\)\(\begin{bmatrix}-3&2&2\\2&-3&2\\2&2&-3\end{bmatrix}\)

\(=\begin{bmatrix}-3+4+4&-6+2+4&-6+4+2\\2-3+4&4-3+4&4-6+2\\2+4-6&4+2-6&4+4-3\end{bmatrix}\)

\(=\begin{bmatrix}5&0&5\\0&5&0\\0&0&5\end{bmatrix}\)

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


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