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$Consider this matrix: [[2,-8],[-2,7]] Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.$

Consider this matrix: $\newline$ $\left[\begin{array}{cc} 2 & -8 \\ -2 & 7 \end{array}\right]$ $\newline$ Find the inverse of the matrix. Give exact values. Non-integers can be given as decimals or as simplified fractions.

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Q. Consider this matrix:

\newline

\left[\begin{array}{cc} 2 & -8 \\ -2 & 7 \end{array}\right]

\newline

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

Write Matrix Determinant: Write down the matrix and its determinant. $\newline$ The matrix is: $\newline$ $A = \left[\begin{array}{cc}2 & -8 \ -2 & 7\end{array}\right]$ $\newline$ To find the inverse of a $2 \times 2$ matrix, we need to calculate its determinant. The determinant of a $2 \times 2$ matrix $A = \left[\begin{array}{cc}a & b \ c & d\end{array}\right]$ is $ad - bc$ . $\newline$ For our matrix $A$ , the determinant ( $\text{det}(A)$ ) is: $\newline$ $\text{det}(A) = (2)(7) - (-8)(-2)$ $\newline$ $\text{det}(A) = 14 - 16$ $\newline$ $\text{det}(A) = -2$
Check Non-Zero Determinant: Check if the determinant is non-zero. $\newline$ Since the determinant is non-zero ( $\text{det}(A) = -2$ ), the matrix has an inverse. If the determinant were zero, the matrix would not have an inverse.
Find Matrix of Minors: Find the matrix of minors. $\newline$ For a $2 \times 2$ matrix, the matrix of minors is simply a matrix where each element is replaced by the determinant of the submatrix formed by removing the row and column of that element. $\newline$ For our matrix $A$ , the matrix of minors is: $\newline$ \left[\begin{array}{cc}$\newline7 & -2 (\newline$-8 & 2 $\newline$ \end{array}\right]\)
Apply Checkerboard of Signs: Apply the checkerboard of signs to the matrix of minors to get the cofactor matrix. $\newline$ For a $2\times2$ matrix, this means we change the sign of the elements at position $(1,2)$ and $(2,1)$ . $\newline$ The cofactor matrix is: $\newline$ \left[\begin{array}{cc}$\newline7 & 2 (\newline$8 & 2 $\newline$ \end{array}\right]\)
Transpose Cofactor Matrix: Transpose the cofactor matrix. $\newline$ The transpose of a matrix is obtained by swapping the rows and columns. $\newline$ The transpose of the cofactor matrix is: $\newline$ $\begin{bmatrix}7 & 8\ 2 & 2\end{bmatrix}$
Multiply by $1/\text{det}(A)$ : Multiply the transpose of the cofactor matrix by $1/\text{det}(A)$ . Since $\text{det}(A) = -2$ , we multiply each element of the transposed cofactor matrix by $-1/2$ to get the inverse of the original matrix. $\newline$ The inverse matrix is: $\newline$ \left[\begin{array}{cc}$\newline-7/2 & -8/2 (\newline$-2/2 & -2/2 $\newline$ \end{array}\right]\)
Simplify Inverse Matrix: Simplify the fractions in the inverse matrix. $\newline$ The simplified inverse matrix is: $\newline$ \left[\begin{array}{cc}$\newline-3.5 & -4 (\newline$-1 & -1 $\newline$ \end{array}\right]\)