To determine the value of [tex]\( a \)[/tex] given that the determinant of the matrix below is [tex]\(-19\)[/tex]:
[tex]\[
\begin{pmatrix}
-6 & 7 & 1 \\
a & -3 & 4 \\
-6 & 4 & -3
\end{pmatrix}
\][/tex]
we need to follow these steps:
1. Construct the Matrix:
Write down the matrix for clarity:
[tex]\[
M = \begin{pmatrix}
-6 & 7 & 1 \\
a & -3 & 4 \\
-6 & 4 & -3
\end{pmatrix}
\][/tex]
2. Calculate the Determinant:
To find the determinant of [tex]\( M \)[/tex], we will use the cofactor expansion along the first row:
[tex]\[
\text{det}(M) = -6 \begin{vmatrix}
-3 & 4 \\
4 & -3
\end{vmatrix}
- 7 \begin{vmatrix}
a & 4 \\
-6 & -3
\end{vmatrix}
+ 1 \begin{vmatrix}
a & -3 \\
-6 & 4
\end{vmatrix}
\][/tex]
3. Evaluate each 2x2 Minor:
Calculate the determinants of each 2x2 minor matrix:
[tex]\[
\det \begin{vmatrix}
-3 & 4 \\
4 & -3
\end{vmatrix} = (-3)(-3) - (4)(4) = 9 - 16 = -7
\][/tex]
[tex]\[
\det \begin{vmatrix}
a & 4 \\
-6 & -3
\end{vmatrix} = (a)(-3) - (4)(-6) = -3a + 24
\][/tex]
[tex]\[
\det \begin{vmatrix}
a & -3 \\
-6 & 4
\end{vmatrix} = (a)(4) - (-3)(-6) = 4a - 18
\][/tex]
4. Substitute these Minors into the Determinant Formula:
Plug these results back into the cofactor expansion:
[tex]\[
\text{det}(M) = -6(-7) - 7(-3a + 24) + 1(4a - 18)
\][/tex]
5. Simplify the Expression:
Simplify this step-by-step:
[tex]\[
\text{det}(M) = 42 + 21a - 168 + 4a - 18
\][/tex]
Combine the like terms:
[tex]\[
\text{det}(M) = 25a - 144
\][/tex]
6. Set the Determinant Equal to -19:
We've been given that:
[tex]\[
25a - 144 = -19
\][/tex]
7. Solve for [tex]\( a \)[/tex]:
Solve the equation for [tex]\( a \)[/tex] step-by-step:
[tex]\[
25a - 144 = -19
\][/tex]
Add 144 to both sides:
[tex]\[
25a = 125
\][/tex]
Divide both sides by 25:
[tex]\[
a = 5
\][/tex]
Therefore, the value of [tex]\( a \)[/tex] is [tex]\( 5 \)[/tex].
