Question

Calculate the dterminant
2 -2 3
4 3 8
5 -1 2

Answer 1

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.

1. Calculate \( ei - fh \):
\[
ei = 3 \cdot 2 = 6
\]
\[
fh = 8 \cdot (-1) = -8
\]
\[
ei - fh = 6 - (-8) = 6 + 8 = 14
\]

2. Calculate \( di - fg \):
\[
di = 4 \cdot 2 = 8
\]
\[
fg = 8 \cdot 5 = 40
\]
\[
di - fg = 8 - 40 = -32
\]

3. 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:

\[
\boxed{-93}
\]