Find the determinant of the matrix \[ A = \left[ \begin{array}{ccc} 2 & 3 & 4 \\ 1 & 5 & 6 \\ 7 & 8 & 9 \end{array} \right] \]
Answer
Step 5: Add the numbers together to find the determinant: \[ \text{det}(A) = 93 \]
Step 1 ：Step 1: Apply the formula for the determinant of a 3x3 matrix: \[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \] where \( a, b, c, d, e, f, g, h, i \) are the elements of the matrix.
Step 2 ：Step 2: Substitute the values from the given matrix into the formula: \[ \text{det}(A) = 2(45 - 48) - 3(6 - 35) + 4(8 - 5) \]
Step 3 ：Step 3: Simplify the expression: \[ \text{det}(A) = 2(-3) - 3(-29) + 4(3) \]
Step 4 ：Step 4: Continue to simplify: \[ \text{det}(A) = -6 + 87 + 12 \]
Step 5 ：Step 5: Add the numbers together to find the determinant: \[ \text{det}(A) = 93 \]
