Find the inverse of the matrix (if it exists) | vedclass

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(C) Let $A = \left[\begin{array}{ccc}2 & 1 & 3 \ 4 & -1 & 0 \ -7 & 2 & 1\end{array}ight]$.First,we calculate the determinant $|A|$:$|A| = 2((-1)(1

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$-\frac{1}{3}\left[\begin{array}{ccc}-1 & 5 & 3 \ -4 & 23 & 12 \ 1 & -11 & -6\end{array}ight]$

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(C) Let $A = \left[\begin{array}{ccc}2 & 1 & 3 \ 4 & -1 & 0 \ -7 & 2 & 1\end{array}ight]$.First,we calculate the determinant $|A|$:$|A| = 2((-1)(1) - (0)(2)) - 1((4)(1) - (0)(-7)) + 3((4)(2) - (-1)(-7))$$|A| = 2(-1 - 0) - 1(4 - 0) + 3(8 - 7)$$|A| = 2(-1) - 1(4) + 3(1) = -2 - 4 + 3 = -3$.Since $|A| 
eq 0$,the inverse $A^{-1}$ exists.Next,we find the cofactor matrix $C_{ij}$:$C_{11} = +((-1)(1) - (0)(2)) = -1$$C_{12} = -((4)(1) - (0)(-7)) = -4$$C_{13} = +((4)(2) - (-1)(-7)) = 8 - 7 = 1$$C_{21} = -((1)(1) - (3)(2)) = -(1 - 6) = 5$$C_{22} = +((2)(1) - (3)(-7)) = 2 + 21 = 23$$C_{23} = -((2)(2) - (1)(-7)) = -(4 + 7) = -11$$C_{31} = +((1)(0) - (3)(-1)) = 0 + 3 = 3$$C_{32} = -((2)(0) - (3)(4)) = -(0 - 12) = 12$$C_{33} = +((2)(-1) - (1)(4)) = -2 - 4 = -6$The adjoint matrix $adj(A)$ is the transpose of the cofactor matrix:$adj(A) = \left[\begin{array}{ccc}-1 & 5 & 3 \ -4 & 23 & 12 \ 1 & -11 & -6\end{array}ight]$.Finally,$A^{-1} = \frac{1}{|A|} adj(A) = \frac{1}{-3} \left[\begin{array}{ccc}-1 & 5 & 3 \ -4 & 23 & 12 \ 1 & -11 & -6\end{array}ight] = -\frac{1}{3} \left[\begin{array}{ccc}-1 & 5 & 3 \ -4 & 23 & 12 \ 1 & -11 & -6\end{array}ight]$.

Find the inverse of the matrix (if it exists): $\left[\begin{array}{ccc}2 & 1 & 3 \\ 4 & -1 & 0 \\ -7 & 2 & 1\end{array}\right]$

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