(a) The determinant of the matrix product AB is π |AB| = -129π.(b) The product of matrices AC is π AC π = beginbmatrix -6 & 18 6 & -19 22 & -109 endbmatrix.(c) The inverse of matrix A, denoted as A⁻¹, does not exist because the determinant of matrix A is zero.(a) To find the determinant of the product of matrices AB, we first calculate the product AB and then find its determinant. Multiplying matrices A and B, we get:AB = beginbmatrix 2 & 4 & -3 1 & 0 & 2 7 & 4 & -10 endbmatrix * beginbmatrix 0 & -3 & 7 3 & -1 & 5 2 & 15 & 8 endbmatrix= beginbmatrix 8 & -9 & 38 4 & -3 & 24 8 & -57 & 114 endbmatrix.Then, calculating the determinant of AB, we get |AB| = -129, and hence, π |AB| π = -129π.(b) To find the product of matrices AC, we simply multiply matrices A and C:AC = beginbmatrix 2 & 4 & -3 1 & 0 & 2 7 & 4 & -10 endbmatrix * beginbmatrix 0 & -3 3 & -1 2 & 15 endbmatrix= beginbmatrix -6 & 18 6 & -19 22 & -109 endbmatrix, yielding π AC π.(c) To find the inverse of matrix A, denoted as A⁻¹, we first check if the determinant of matrix A is non-zero. However, the determinant of A is zero (|A| = 0), indicating that the matrix is singular, and hence, its inverse does not exist. Therefore, π A⁻¹ π is undefined....

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