Question - Matrix Determinant Calculation
Solution:
The provided photo seems to show the determinant of a 3x3 matrix being calculated. However, the step-by-step calculation is not fully visible, and there's some text in another language at the bottom. With the visible content, I will assume the goal is to calculate the determinant of a matrix displayed at the top right corner of the image.
Let the matrix A be:
\[ A = \begin{bmatrix} -3 & 1 & 5 \\ 1 & -3 & -5 \\ 1 & 3 & 5 \\ \end{bmatrix} \]To find the determinant of matrix A (denoted as det(A) or |A|), we can use the following formula for a 3x3 matrix:
det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
Substituting the values from matrix A, we get:
\[ \begin{align*} \text{det}(A) &= (-3)((-3)(5) - (-5)(3)) - (1)((1)(5) - (3)(-5)) + (5)((1)(3) - (-3)(1)) \\ &= (-3)((-15) - (-15)) - (1)((5) - (-15)) + (5)((3) - (-3)) \\ &= (-3)(0) - (1)(20) + (5)(6) \\ &= 0 - 20 + 30 \\ &= 10 \end{align*} \]Therefore, the determinant of matrix A is 10.
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