To calculate the determinant of the given $3 \times 3$ matrix:

$\begin{vmatrix} 5 & 2 & -2 \\ 7 & 8 & -2 \\ -8 & -5 & 6 \end{vmatrix}$

we can use the formula for the determinant of a $3 \times 3$ matrix:

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

where the matrix elements are as follows:

$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$

In our case:

• $a = 5$, $b = 2$, $c = -2$
• $d = 7$, $e = 8$, $f = -2$
• $g = -8$, $h = -5$, $i = 6$

Step-by-Step Calculation:

1. Calculate $a(ei - fh)$: $5 \times (8 \times 6 - (-2) \times (-5)) = 5 \times (48 - 10) = 5 \times 38 = 190$

2. Calculate $-b(di - fg)$: $-2 \times (7 \times 6 - (-2) \times (-8)) = -2 \times (42 - 16) = -2 \times 26 = -52$

3. Calculate $c(dh - eg)$: $-2 \times (7 \times -5 - 8 \times -8) = -2 \times (-35 + 64) = -2 \times 29 = -58$

4. Add the results: $190 - 52 - 58 = 80$

Final Answer:

$\text{det}(A) = 80$

Would you like more details on the steps?

Related Questions:

1. How do you calculate determinants of larger matrices?
2. What is the significance of the determinant in linear algebra?
3. How does the determinant relate to matrix invertibility?
4. Can determinants be used to find the area of shapes in geometry?
5. What are some properties of determinants that simplify calculations?

Tip:

When calculating determinants, always check for possible row or column simplifications to reduce computation complexity.