Full solution

Q. Consider this matrix:

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\left[\begin{array}{cc} -4 & -8 \\ -5 & -5 \end{array}\right]

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Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

Calculate Determinant: To find the inverse of a $2$ x $2$ matrix, we use the formula: $\newline$ Inverse(A) = $(1/\text{det}(A)) \cdot \text{adj}(A)$ $\newline$ where $\text{det}(A)$ is the determinant of matrix A and $\text{adj}(A)$ is the adjugate of matrix A. $\newline$ First, we need to calculate the determinant of the matrix. $\newline$ $\text{det}(A) = (-4)(-5) - (-8)(-5)$ $\newline$ $\text{det}(A) = 20 - 40$ $\newline$ $\text{det}(A) = -20$
Find Adjugate: Since the determinant is not zero, the matrix has an inverse. Now we need to find the adjugate of the matrix. $\newline$ The adjugate of a $2 \times 2$ matrix is obtained by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. $\newline$ So, the adjugate of matrix $A$ is: $\newline$ $\text{adj}(A) = \left[\begin{array}{cc}-5 & 8 \ 5 & -4\end{array}\right]$
Calculate Inverse: Now we can find the inverse of the matrix by multiplying the adjugate by $1/\det(A)$ . $\newline$ Inverse( $A$ ) = $(1/-20) \times \left[\begin{array}{cc} -5 & 8 \ 5 & -4 \end{array}\right]$ $\newline$ Inverse( $A$ ) = $\left[\begin{array}{cc} -5/-20 & 8/-20 \ 5/-20 & -4/-20 \end{array}\right]$ $\newline$ Inverse( $A$ ) = $\left[\begin{array}{cc} 1/4 & -2/5 \ -1/4 & 1/5 \end{array}\right]$