To calculate the determinant of the given 3x3 matrix:
\[ \begin{bmatrix} 2 & -2 & 3 \ 4 & 3 & 8 \ 5 & -1 & 2 \end{bmatrix} \]
We can use the formula for the determinant of a 3x3 matrix:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix} \]
In our case:
- \( a = 2 \), \( b = -2 \), \( c = 3 \)
- \( d = 4 \), \( e = 3 \), \( f = 8 \)
- \( g = 5 \), \( h = -1 \), \( i = 2 \)
Now we can compute the determinant step by step.
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Calculate \( ei - fh \): \[ ei = 3 \cdot 2 = 6 \] \[ fh = 8 \cdot (-1) = -8 \] \[ ei - fh = 6 - (-8) = 6 + 8 = 14 \]
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Calculate \( di - fg \): \[ di = 4 \cdot 2 = 8 \] \[ fg = 8 \cdot 5 = 40 \] \[ di - fg = 8 - 40 = -32 \]
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Calculate \( dh - eg \): \[ dh = 4 \cdot (-1) = -4 \] \[ eg = 3 \cdot 5 = 15 \] \[ dh - eg = -4 - 15 = -19 \]
Putting all this back into the determinant formula:
\[ \text{det}(A) = 2(14) - (-2)(-32) + 3(-19) \]
Calculating each term:
- \( 2(14) = 28 \)
- \( -(-2)(-32) = -64 \)
- \( 3(-19) = -57 \)
Now sum these values:
\[ \text{det}(A) = 28 - 64 - 57 \] \[ = 28 - 121 \] \[ = -93 \]
Thus, the determinant of the matrix is:
\[ \boxed{-93} \]
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