To find the determinant of the matrix [tex]\( L = \begin{bmatrix} -5 & -5 & 3 \\ 5 & -3 & 2 \\ 3 & 1 & 2 \end{bmatrix} \)[/tex], we follow these steps:
1. Set up the determinant formula for a 3x3 matrix:
[tex]\[ \text{det}(L) = \begin{vmatrix} -5 & -5 & 3 \\ 5 & -3 & 2 \\ 3 & 1 & 2 \end{vmatrix}. \][/tex]
The determinant of a 3x3 matrix [tex]\( \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \)[/tex] is calculated as follows:
[tex]\[ \text{det}(L) = a(ei - fh) - b(di - fg) + c(dh - eg). \][/tex]
2. Assign the matrix elements to the corresponding variables:
[tex]\[ a = -5, \quad b = -5, \quad c = 3 \][/tex]
[tex]\[ d = 5, \quad e = -3, \quad f = 2 \][/tex]
[tex]\[ g = 3, \quad h = 1, \quad i = 2 \][/tex]
3. Substitute the elements into the determinant formula:
[tex]\[ \text{det}(L) = (-5)[(-3)(2) - (2)(1)] - (-5)[(5)(2) - (2)(3)] + 3[(5)(1) - (-3)(3)]. \][/tex]
4. Evaluate the products inside the parentheses:
[tex]\[ (-3)(2) = -6 \][/tex]
[tex]\[ (2)(1) = 2 \][/tex]
[tex]\[ (5)(2) = 10 \][/tex]
[tex]\[ (2)(3) = 6 \][/tex]
[tex]\[ (5)(1) = 5 \][/tex]
[tex]\[ (-3)(3) = -9 \][/tex]
5. Perform the arithmetic operations inside the parentheses:
[tex]\[ (-3)(2) - (2)(1) = -6 - 2 = -8 \][/tex]
[tex]\[ (5)(2) - (2)(3) = 10 - 6 = 4 \][/tex]
[tex]\[ (5)(1) - (-3)(3) = 5 - (-9) = 5 + 9 = 14 \][/tex]
6. Substitute back into the determinant expression and perform the final arithmetic:
[tex]\[ \text{det}(L) = (-5)(-8) - (-5)(4) + 3(14) \][/tex]
[tex]\[ = 40 - (-20) + 42 \][/tex]
[tex]\[ = 40 + 20 + 42 \][/tex]
[tex]\[ = 102. \][/tex]
Therefore, the determinant of the matrix [tex]\( L \)[/tex] is [tex]\( 102 \)[/tex].
