To find the value of [tex]\( a \)[/tex] such that the determinant of the given matrix is [tex]\(-19\)[/tex], we need to compute the determinant and solve for [tex]\( a \)[/tex]. The matrix is:
[tex]\[
\left[\begin{array}{ccc}
-6 & 7 & 1 \\
a & -3 & 4 \\
-6 & 4 & -3
\end{array}\right]
\][/tex]
The determinant of a [tex]\(3 \times 3\)[/tex] matrix [tex]\( \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \)[/tex] is given by:
[tex]\[
\text{det} = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})
\][/tex]
Let's substitute the given matrix elements into this formula:
[tex]\[
\begin{array}{l}
a_{11} = -6, \quad a_{12} = 7, \quad a_{13} = 1, \\
a_{21} = a, \quad a_{22} = -3, \quad a_{23} = 4, \\
a_{31} = -6, \quad a_{32} = 4, \quad a_{33} = -3
\end{array}
\][/tex]
Now, compute the determinant step by step:
[tex]\[
\begin{vmatrix}
-6 & 7 & 1 \\
a & -3 & 4 \\
-6 & 4 & -3
\end{vmatrix}
= -6 [(-3)(-3) - (4)(4)] - 7 [(a)(-3) - (4)(-6)] + 1 [(a)(4) - (-3)(-6)]
\][/tex]
Compute the individual terms within the brackets:
[tex]\[
-6 [(9) - (16)]
= -6 (-7)
= 42
\][/tex]
For the second term:
[tex]\[
-7 [(a)(-3) - (4)(-6)]
= -7 [-3a + 24]
= -7(-3a + 24)
= 21a - 168
\][/tex]
For the third term:
[tex]\[
1 [(4a) - (9)]
= 4a - 9
\][/tex]
Combine all the terms:
[tex]\[
\text{det} = 42 + 21a - 168 + 4a - 9
\][/tex]
Simplify this expression:
[tex]\[
\text{det} = 21a + 4a + 42 - 168 - 9
= 25a - 135
\][/tex]
We are given that the determinant is [tex]\(-19\)[/tex]. Set the determinant equation equal to [tex]\(-19\)[/tex] and solve for [tex]\( a \)[/tex]:
[tex]\[
25a - 135 = -19
\][/tex]
Add 135 to both sides:
[tex]\[
25a = 116
\][/tex]
Divide by 25:
[tex]\[
a = \frac{116}{25}
= 4.64
\][/tex]
Since none of the options A, B, C, or D match exactly, there might be an error in the process of solving or in the provided problem options. However, according to our solution, the calculated value of [tex]\( a \)[/tex] is approximately [tex]\( 4.64 \)[/tex].
