1.

Find the inverse of the following matrices by the adjoint method :\( \begin{bmatrix}2&-2 \\[0.3em]4 & 3 \\[0.3em] \end{bmatrix} \)[2,-2,4,3]

Answer»

Let A = [2,-2,4,3]

\( \begin{bmatrix}2&-2 \\[0.3em]4 & 3 \\[0.3em] \end{bmatrix} \)

|A| = = 6 + 8 = 14 ≠ 0 

∴ A-1 exist 

First we have to find the co-factor matrix

= [Aij]2x2 where Aij = (-1)i+jMij

Now, 

A11 = (-1)1+1 M11 = 3 

A12 = (-1)1+2M = -4 

A21 = (-2)2+1M21 = (-2) = 2 

A22 = (-1)2+2M22 = 2 

Hence the co-factor matrix

= \( \begin{bmatrix}A_{11}&A_{12} \\[0.3em]A_{21} & A_{22} \\[0.3em] \end{bmatrix} \)

= \( \begin{bmatrix}3&-4 \\[0.3em]2 & 2 \\[0.3em] \end{bmatrix} \)

∴ adj A = \( \begin{bmatrix}3&2 \\[0.3em]-4& 2 \\[0.3em] \end{bmatrix} \)

∴ A-1 = \(\frac{1}{|A|}\) (adj A)

= \(\frac{1}{14}\)\( \begin{pmatrix}3&2 \\[0.3em]-4& 2 \\[0.3em] \end{pmatrix} \)


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