Questions: Find the determinant of the matrix: 12 -4 0 -8 8 0 4 32 4 -512 -448 0 256 192

Solution Steps

To find the determinant of a 3x3 matrix, we can use the rule of Sarrus or the general formula for determinants. The determinant of a 3x3 matrix \(\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\) is calculated as \(a(ei - fh) - b(di - fg) + c(dh - eg)\).

To find the determinant of the matrix

\[ \begin{bmatrix} 12 & -4 & 0 \\ -8 & 8 & 0 \\ 4 & 32 & 4 \end{bmatrix} \]

we use the formula for the determinant of a 3x3 matrix:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

where \( a = 12, b = -4, c = 0, d = -8, e = 8, f = 0, g = 4, h = 32, i = 4 \).

Substituting the values into the determinant formula, we have:

\[ \text{det}(A) = 12(8 \cdot 4 - 0 \cdot 32) - (-4)(-8 \cdot 4 - 0 \cdot 4) + 0(-8 \cdot 32 - 8 \cdot 4) \]

Calculating each term:

1. \( ei - fh = 8 \cdot 4 - 0 \cdot 32 = 32 \)
2. \( di - fg = -8 \cdot 4 - 0 \cdot 4 = -32 \)
3. The third term is zero since \( c = 0 \).

Thus, we have:

\[ \text{det}(A) = 12 \cdot 32 - (-4)(-32) + 0 = 384 - 128 + 0 = 256 \]

Final Answer