Given the matrix: A = [[2, 1, 3], [0, -1, 2], [4, 5, 1]]. 2.2.1 Calculate the determinant of matrix A. 2.2.2 Determine the cofactor of element a23.

Here are the solutions for the matrix problems: Given the matrix: A = 2 & 1 & 3 \\ 0 & -1 & 2 \\ 4 & 5 & 1 2.2.1 Calculate the determinant of matrix A. Step 1: Use the cofactor expansion method. We will expand along the first row. The formula for the determinant of a 3 × 3 matrix is (A) = a_11C_11 + a_12C_12 + a_13C_13, where C_ij = (-1)^i+jM_ij. (A) = 2 · (-1)^1+1 -1 & 2 \\ 5 & 1 + 1 · (-1)^1+2 0 & 2 \\ 4 & 1 + 3 · (-1)^1+3 0 & -1 \\ 4 & 5 Step 2: Calculate the 2 × 2 determinants (minors). -1 & 2 \\ 5 & 1 = (-1)(1) - (2)(5) = -1 - 10 = -11 0 & 2 \\ 4 & 1 = (0)(1) - (2)(4) = 0 - 8 = -8 0 & -1 \\ 4 & 5 = (0)(5) - (-1)(4) = 0 - (-4) = 4 Step 3: Substitute the minor values back into the determinant formula and simplify. (A) = 2(1)(-11) + 1(-1)(-8) + 3(1)(4) (A) = -22 + 8 + 12 (A) = -22 + 20 (A) = -2 The determinant of matrix A is -2. 2.2.2 Determine the cofactor of element a_23. Step 1: Identify the element a_23. It is the element in the 2nd row and 3rd column, which is 2. The formula for the cofactor C_ij is C_ij = (-1)^i+jM_ij, where M_ij is the minor. For a_23, we need to find C_23. Step 2: Determine the minor M_23 by deleting the 2nd row and 3rd column of matrix A. M_23 = 2 & 1 \\ 4 & 5 Step 3: Calculate the determinant of the 2 × 2 minor matrix. M_23 = (2)(5) - (1)(4) = 10 - 4 = 6 Step 4: Calculate the cofactor C_23. C_23 = (-1)^2+3 M_23 C_23 = (-1)^5 (6) C_23 = -1 · 6 C_23 = -6 The cofactor of element a_23 is -6. 3 done, 2 left today. You're making progress.