Answer:
-7
Step-by-step explanation:
To find the determinant of the matrix [tex]H[/tex], denoted as [tex]|H|[/tex], we can use the method of expansion by minors or cofactor expansion.
Given the matrix:
[tex] H = \begin{bmatrix} 0 & 2 & 3 \\ -1 & 3 & 5 \\ 6 & 3 & -2 \end{bmatrix} [/tex]
We can expand along the first row:
[tex] |H| = 0 \times \begin{vmatrix} 3 & 5 \\ 3 & -2 \end{vmatrix} - 2 \times \begin{vmatrix} -1 & 5 \\ 6 & -2 \end{vmatrix} + 3 \times \begin{vmatrix} -1 & 3 \\ 6 & 3 \end{vmatrix} [/tex]
Now, let's compute the determinants of each 2×2 matrix:
[tex] \begin{vmatrix} 3 & 5 \\ 3 & -2 \end{vmatrix} = (3 \times -2) - (5 \times 3) = -6 - 15 = -21 [/tex]
[tex] \begin{vmatrix} -1 & 5 \\ 6 & -2 \end{vmatrix} = (-1 \times -2) - (5 \times 6) = 2 - 30 = -28 [/tex]
[tex] \begin{vmatrix} -1 & 3 \\ 6 & 3 \end{vmatrix} = (-1 \times 3) - (3 \times 6) = -3 - 18 = -21 [/tex]
Now, substitute these values into the expression for [tex]|H|[/tex]:
[tex] |H| = 0 \times (-21) - 2 \times (-28) + 3 \times (-21) [/tex]
[tex] |H| = 0 + 56 - 63 [/tex]
[tex] |H| = -7 [/tex]
Therefore, the determinant of matrix [tex]H[/tex] is [tex]-7[/tex].
