Let X = bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix and I be the 3 3 identity matrix. If a matrix Y is defined as Y = I + X for some real numbers and , and its inverse is given by Y⁻¹ = 3I - X , then the value of 18 + 36 is equal to ______.

Step-by-step solution

First, we compute X^2 to check for any special properties of the matrix X . X^2 = bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix bmatrix 2 & -2 & -4 -1 & 3 & 4 1 & -2 & -3 bmatrix Calculating the elements of X^2 : Row 1, Col 1: 2(2) + (-2)(-1) + (-4)(1) = 4 + 2 - 4 = 2 Row 1, Col 2: 2(-2) + (-2)(3) + (-4)(-2) = -4 - 6 + 8 = -2 Row 1, Col 3: 2(-4) + (-2)(4) + (-4)(-3) = -8 - 8 + 12 = -4 Row 2, Col 1: -1(2) + 3(-1) + 4(1) = -2 - 3 + 4 = -1 Row 2, Col 2: -1(-2) + 3(3) + 4(-2) = 2 + 9 - 8 = 3 Row 2, Col 3: -1(-4) +