Solve the determinant equation:

[tex]\[ \left|\begin{array}{ccc}
7 & 2 & -1 \\
3x & 2 & 2 \\
-3 & 1 & -7
\end{array}\right| = 65 \][/tex]



Answer :

To solve the given determinant equation for [tex]\( x \)[/tex]:

[tex]\[ \left|\begin{array}{ccc} 7 & 2 & -1 \\ 3x & 2 & 2 \\ -3 & 1 & -7 \end{array}\right| = 65 \][/tex]

we need to follow these steps:

1. Calculate the Determinant of the Matrix:

We start by calculating the determinant of the [tex]\( 3 \times 3 \)[/tex] matrix:

[tex]\[ A = \begin{vmatrix} 7 & 2 & -1 \\ 3x & 2 & 2 \\ -3 & 1 & -7 \end{vmatrix} \][/tex]

Using the rule for finding the determinant of a [tex]\( 3 \times 3 \)[/tex] matrix, we have:

[tex]\[ \text{det}(A) = 7 \begin{vmatrix} 2 & 2 \\ 1 & -7 \end{vmatrix} - 2 \begin{vmatrix} 3x & 2 \\ -3 & -7 \end{vmatrix} - 1 \begin{vmatrix} 3x & 2 \\ -3 & 1 \end{vmatrix} \][/tex]

Now, calculate the [tex]\( 2 \times 2 \)[/tex] determinants:

[tex]\[ \begin{vmatrix} 2 & 2 \\ 1 & -7 \end{vmatrix} = (2)(-7) - (2)(1) = -14 - 2 = -16 \][/tex]

[tex]\[ \begin{vmatrix} 3x & 2 \\ -3 & -7 \end{vmatrix} = (3x)(-7) - (2)(-3) = -21x + 6 \][/tex]

[tex]\[ \begin{vmatrix} 3x & 2 \\ -3 & 1 \end{vmatrix} = (3x)(1) - (2)(-3) = 3x + 6 \][/tex]

Substitute these back into the determinant formula:

[tex]\[ \text{det}(A) = 7(-16) - 2(-21x + 6) - 1(3x + 6) \][/tex]

Simplify:

[tex]\[ \text{det}(A) = -112 + 42x - 12 - 3x - 6 \][/tex]

Combine like terms:

[tex]\[ \text{det}(A) = 39x - 130 \][/tex]

2. Set the Determinant Equal to 65:

We are given that the determinant is equal to 65. So, we set up the equation:

[tex]\[ 39x - 130 = 65 \][/tex]

3. Solve for [tex]\( x \)[/tex]:

To isolate [tex]\( x \)[/tex], add 130 to both sides of the equation:

[tex]\[ 39x = 195 \][/tex]

Now, divide by 39:

[tex]\[ x = \frac{195}{39} = 5 \][/tex]

Thus, the solution to the equation is:

[tex]\[ x = 5 \][/tex]

So, we have calculated that the determinant of the matrix equals [tex]\( 39x - 130 \)[/tex], and solving for [tex]\( x \)[/tex] when this determinant is set to 65, we find:

[tex]\[ x = 5 \][/tex]