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Q : 10         Find the inverse of each of the matrices (if it exists).

                   \small \begin{bmatrix} 1 & -1 & 2\\ 0 & 2 &-3 \\ 3 &-2 &4 \end{bmatrix}

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Given the matrix :  \small \begin{bmatrix} 1 & -1 & 2\\ 0 & 2 &-3 \\ 3 &-2 &4 \end{bmatrix} = A

To find the inverse we have to first find adjA then as we know the relation:

A^{-1} = \frac{1}{|A|}adjA

So, calculating |A| :

|A| = 1(8-6)+1(0+9)+2(0-6) =2+9-12 = -1

Now, calculating the cofactors terms and then adjA.

A_{11} = (-1)^{1+1} (8-6) = 2                  A_{12} = (-1)^{1+2} (0+9) = -9

A_{13} = (-1)^{1+3} (0-6) =-6                      A_{21} = (-1)^{2+1} (-4+4) = 0

A_{22} = (-1)^{2+2} (4-6) = -2             A_{23} = (-1)^{2+1} (-2+3) = -1

A_{31} = (-1)^{3+1} (3-4) = -1             A_{32} = (-1)^{3+2} (-3-0) =3

A_{33} = (-1)^{3+3} (2-0) = 2               

So, we have adjA = \begin{bmatrix} 2 &0 &-1 \\ -9& -2 &3 \\ -6& -1 &2 \end{bmatrix}

Therefore inverse of A will be:

A^{-1} = \frac{1}{|A|}adjA

A^{-1} = \frac{1}{-1} \begin{bmatrix} 2 &0 &-1 \\ -9& -2 &3 \\ -6& -1 &2 \end{bmatrix}

A^{-1} = \begin{bmatrix} -2 &0 &1 \\ 9& 2 &-3 \\ 6& 1 &-2 \end{bmatrix}

Posted by

Divya Prakash Singh

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