To solve the problem:
[tex]\[
\left|\begin{array}{rrr}
x & -1 & 2 \\
2x & 1 & -3 \\
3 & -4 & 5
\end{array}\right| = 29
\][/tex]
we need to calculate the determinant of the matrix and solve for [tex]\( x \)[/tex] when the determinant equals 29. Here is the step-by-step solution:
1. Define the Matrix:
[tex]\[
A = \begin{pmatrix}
x & -1 & 2 \\
2x & 1 & -3 \\
3 & -4 & 5
\end{pmatrix}
\][/tex]
2. Calculate the Determinant:
The determinant of a 3x3 matrix can be computed using the general 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 our matrix [tex]\( A \)[/tex]:
[tex]\[
a = x, \quad b = -1, \quad c = 2
\][/tex]
[tex]\[
d = 2x, \quad e = 1, \quad f = -3
\][/tex]
[tex]\[
g = 3, \quad h = -4, \quad i = 5
\][/tex]
Plug these values into the determinant formula:
[tex]\[
\text{det}(A) = x \left(1 \cdot 5 - (-3) \cdot (-4)\right) - (-1) \left(2x \cdot 5 - (-3) \cdot 3\right) + 2 \left(2x \cdot (-4) - 1 \cdot 3\right)
\][/tex]
Simplify each term:
[tex]\[
= x (5 - 12) + (2x \cdot 5 + 9) + 2 (-8x - 3)
= x (-7) + (10x + 9) + 2 (-8x - 3)
= -7x + 10x + 9 - 16x - 6
= -7x + 10x - 16x + 9 - 6
= -13x + 3
\][/tex]
So the determinant of the matrix is:
[tex]\[
\text{det}(A) = 3 - 13x
\][/tex]
3. Set the Determinant Equal to 29 and Solve for [tex]\( x \)[/tex]:
[tex]\[
3 - 13x = 29
\][/tex]
Solve the equation for [tex]\( x \)[/tex]:
[tex]\[
-13x = 29 - 3
\][/tex]
[tex]\[
-13x = 26
\][/tex]
[tex]\[
x = \frac{26}{-13}
\][/tex]
[tex]\[
x = -2
\][/tex]
Thus, the value of [tex]\( x \)[/tex] that satisfies the determinant equation is:
[tex]\[
x = -2
\][/tex]
So the determinant of the matrix is [tex]\( 3 - 13x \)[/tex], and the solution to the equation when the determinant equals 29 is [tex]\( x = -2 \)[/tex].
