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The inverse of the matrix $\begin{bmatrix} 2 & 3 & 4\\ 4 & 3 & 1\\ 1 & 2 & 4 \end{bmatrix}$ is

1. $\begin{bmatrix} 10 & -4 & -9\\ -15 & 4 & 14\\ 5 & -1 & -6 \end{bmatrix}$
2. $\begin{bmatrix} -10 & 4 & 9\\ 15 & -4 & -14\\ -5 & 1 & 6 \end{bmatrix}$
3. $\begin{bmatrix} -2 & \frac{4}{5} & \frac{9}{5}\\ 3 & -\frac{4}{5} & \frac{14}{5}\\ -1 & \frac{1}{5} & \frac{6}{5} \end{bmatrix}$
4. $\begin{bmatrix} 2 & -\frac{4}{5} & -\frac{9}{5}\\ -3 & \frac{4}{5} & \frac{14}{5}\\ 1 & -\frac{1}{5} & -\frac{6}{5} \end{bmatrix}$

Correct answer: $\begin{bmatrix} -2 & \frac{4}{5} & \frac{9}{5}\\ 3 & -\frac{4}{5} & \frac{14}{5}\\ -1 & \frac{1}{5} & \frac{6}{5} \end{bmatrix}$

Solution

The inverse of a matrix is found using $A^{-1}=\frac{1}{|A|}\operatorname{adj}(A)$. For this matrix, the determinant is 5, so the inverse must have entries divided by 5, matching the given option.

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