To solve the equation given as the determinant of a 3x3 matrix equal to zero, we will proceed by calculating the determinant of the matrix. The matrix in question is:
[tex]\[
\begin{pmatrix}
3 & 4 & -1 \\
2 & 0 & 7 \\
1 & -3 & -2
\end{pmatrix}
\][/tex]
To find the determinant of a 3x3 matrix, we use the standard formula:
[tex]\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
where
[tex]\[
A =
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\][/tex]
For the given matrix:
[tex]\[
\begin{pmatrix}
3 & 4 & -1 \\
2 & 0 & 7 \\
1 & -3 & -2
\end{pmatrix}
\][/tex]
We can assign the values as follows:
[tex]\[
a = 3, \; b = 4, \; c = -1, \; d = 2, \; e = 0, \; f = 7, \; g = 1, \; h = -3, \; i = -2
\][/tex]
Substituting these values into the formula:
[tex]\[
\text{det}(A) = 3((0 \cdot -2) - (7 \cdot -3)) - 4((2 \cdot -2) - (7 \cdot 1)) + (-1)((2 \cdot -3) - (0 \cdot 1))
\][/tex]
Now calculate each term individually:
1. Calculate [tex]\(3((0 \cdot -2) - (7 \cdot -3))\)[/tex]:
[tex]\[
3(0 + 21) = 3 \cdot 21 = 63
\][/tex]
2. Calculate [tex]\(-4((2 \cdot -2) - (7 \cdot 1))\)[/tex]:
[tex]\[
-4(-4 - 7) = -4(-11) = 44
\][/tex]
3. Calculate [tex]\((-1)((2 \cdot -3) - (0 \cdot 1))\)[/tex]:
[tex]\[
-1(-6 - 0) = -1(-6) = 6
\][/tex]
Sum these results:
[tex]\[
\text{det}(A) = 63 + 44 + 6 = 113
\][/tex]
Thus, the determinant of the matrix is 113. Given that the determinant is not equal to zero, the original equation should be interpreted based on correct determinant properties. The determinant of the given matrix is indeed 113.
