This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the solution to the problems: (d) Show that the matrix M is invertible and find M^-1 Given the matrix: M = 3 & -2 & 5 \\ 7 & 4 & -8 \\ 5 & -3 & -4 Step 1: Calculate the determinant of M. A matrix is invertible if and only if its determinant is non-zero. (M) = 3 4 & -8 \\ -3 & -4 - (-2) 7 & -8 \\ 5 & -4 + 5 7 & 4 \\ 5 & -3 (M) = 3((4)(-4) - (-8)(-3)) + 2((7)(-4) - (-8)(5)) + 5((7)(-3) - (4)(5)) (M) = 3(-16 - 24) + 2(-28 - (-40)) + 5(-21 - 20) (M) = 3(-40) + 2(-28 + 40) + 5(-41) (M) = -120 + 2(12) - 205 (M) = -120 + 24 - 205 (M) = -96 - 205 (M) = -301 Since (M) = -301 ≠ 0, the matrix M is invertible. Step 2: Calculate the cofactor matrix C. C_11 = 4 & -8 \\ -3 & -4 = -16 - 24 = -40 C_12 = - 7 & -8 \\ 5 & -4 = -(-28 + 40) = -12 C_13 = 7 & 4 \\ 5 & -3 = -21 - 20 = -41 C_21 = - -2 & 5 \\ -3 & -4 = -(8 + 15) = -