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(D) We know that the inverse of a matrix $A$ is given by the formula $A^{-1} = \frac{1}{|A|} 	ext{adj}(A)$.Comparing this with the given equation $A^

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$11$

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(D) We know that the inverse of a matrix $A$ is given by the formula $A^{-1} = \frac{1}{|A|} 	ext{adj}(A)$.Comparing this with the given equation $A^{-1} = \frac{1}{K} 	ext{adj}(A)$,we get $K = |A|$.Now,we calculate the determinant of matrix $A$:$|A| = \begin{vmatrix} 3 & 2 & 4 \ 1 & 2 & -1 \ 0 & 1 & 1 \end{vmatrix}$Expanding along the first column:$|A| = 3(2(1) - (-1)(1)) - 1(2(1) - 4(1)) + 0(2(-1) - 4(2))$$|A| = 3(2 + 1) - 1(2 - 4) + 0$$|A| = 3(3) - 1(-2)$$|A| = 9 + 2 = 11$.Therefore,$K = 11$.

If matrix $A = \begin{bmatrix} 3 & 2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{K} \text{adj}(A)$,then $K$ is:

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