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.

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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.

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Write Matrix Determinant: Write down the matrix and its determinant. The matrix is: A = [[2,-8],[-2,7]]  To find the inverse of a 2x2 matrix, we need to calculate its determinant. The determinant of a 2x2 matrix A = [[a, b], [c, d]] is ad - bc.  For our matrix A, the determinant (det(A)) is: det(A) = (2)(7) - (-8)(-2) det(A) = 14 - 16 det(A) = -2Check Non-Zero Determinant: Check if the determinant is non-zero. Since the determinant is non-zero (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. For a 2x2 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.  For our matrix A, the matrix of minors is: [[7, -2],  [-8, 2]]Apply Checkerboard of Signs: Apply the checkerboard of signs to the matrix of minors to get the cofactor matrix. For a 2x2 matrix, this means we change the sign of the elements at position (1,2) and (2,1).  The cofactor matrix is: [[7, 2],  [8, 2]]Transpose Cofactor Matrix: Transpose the cofactor matrix. The transpose of a matrix is obtained by swapping the rows and columns.  The transpose of the cofactor matrix is: [[7, 8],  [2, 2]]Multiply by 1/det(A): Multiply the transpose of the cofactor matrix by 1/det(A). Since det(A) = -2, we multiply each element of the transposed cofactor matrix by -1/2 to get the inverse of the original matrix.  The inverse matrix is: [[-7/2, -8/2],  [-2/2, -2/2]]Simplify Inverse Matrix: Simplify the fractions in the inverse matrix. The simplified inverse matrix is: [[-3.5, -4],  [-1, -1]]

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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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.