Answer :
Let's find the value of [tex]\( x \)[/tex] in the given expression:
[tex]\[
x = \frac{\left| \begin{array}{rrr} -5 & 2 & -3 \\ -6 & -3 & 6 \\ 15 & -6 & 9 \end{array}\right|}
{\left| \begin{array}{rrr} -3 & 2 & -3 \\ 4 & -3 & 6 \\ 9 & -6 & 9 \end{array} \right|}
\][/tex]
First, we need to find the determinant of the numerator matrix:
[tex]\[
\left| \begin{array}{rrr} -5 & 2 & -3 \\ -6 & -3 & 6 \\ 15 & -6 & 9 \end{array} \right|
\][/tex]
To find the determinant of a 3x3 matrix, we use the following formula:
[tex]\[
\left| \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right| = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
Applying this to our numerator matrix:
[tex]\[
= -5 \left( (-3)(9) - (6)(-6) \right) - 2 \left( (-6)(9) - (6)(15) \right) + (-3) \left( (-6)(-6) - (-3)(15) \right)
\][/tex]
Calculate each term inside:
[tex]\[
-5 \left( -27 + 36 \right) - 2 \left( -54 - 90 \right) - 3 \left( 36 - (-45) \right)
\][/tex]
[tex]\[
= -5 \left( 9 \right) - 2 \left( -144 \right) - 3 \left( 81 \right)
\][/tex]
[tex]\[
= -45 + 288 - 243
\][/tex]
[tex]\[
= 0
\][/tex]
So the determinant of the numerator matrix is 0.
Next, we find the determinant of the denominator matrix:
[tex]\[
\left| \begin{array}{rrr} -3 & 2 & -3 \\ 4 & -3 & 6 \\ 9 & -6 & 9 \end{array} \right|
\][/tex]
Using the same formula:
[tex]\[
= -3 \left( (-3)(9) - (6)(-6) \right) - 2 \left( 4(9) - (6)(9) \right) + (-3) \left( 4(-6) - (-3)(9) \right)
\][/tex]
Calculate each term inside:
[tex]\[
-3 \left( -27 + 36 \right) - 2 \left( 36 - 54 \right) - 3 \left( -24 + 27 \right)
\][/tex]
[tex]\[
= -3 \left( 9 \right) - 2 \left( -18 \right) - 3 \left( 3 \right)
\][/tex]
[tex]\[
= -27 + 36 - 9
\][/tex]
[tex]\[
= 0
\][/tex]
So the determinant of the denominator matrix is 0 as well.
Since both determinants are 0, the expression we need to evaluate is:
[tex]\[
x = \frac{0}{0}
\][/tex]
This results in an undefined form. In mathematical terms, the value of [tex]\( x \)[/tex] is "not a number" (NaN), which means that [tex]\( x \)[/tex] cannot be determined from the given matrices.
[tex]\[
x = \frac{\left| \begin{array}{rrr} -5 & 2 & -3 \\ -6 & -3 & 6 \\ 15 & -6 & 9 \end{array}\right|}
{\left| \begin{array}{rrr} -3 & 2 & -3 \\ 4 & -3 & 6 \\ 9 & -6 & 9 \end{array} \right|}
\][/tex]
First, we need to find the determinant of the numerator matrix:
[tex]\[
\left| \begin{array}{rrr} -5 & 2 & -3 \\ -6 & -3 & 6 \\ 15 & -6 & 9 \end{array} \right|
\][/tex]
To find the determinant of a 3x3 matrix, we use the following formula:
[tex]\[
\left| \begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i \end{array} \right| = a(ei - fh) - b(di - fg) + c(dh - eg)
\][/tex]
Applying this to our numerator matrix:
[tex]\[
= -5 \left( (-3)(9) - (6)(-6) \right) - 2 \left( (-6)(9) - (6)(15) \right) + (-3) \left( (-6)(-6) - (-3)(15) \right)
\][/tex]
Calculate each term inside:
[tex]\[
-5 \left( -27 + 36 \right) - 2 \left( -54 - 90 \right) - 3 \left( 36 - (-45) \right)
\][/tex]
[tex]\[
= -5 \left( 9 \right) - 2 \left( -144 \right) - 3 \left( 81 \right)
\][/tex]
[tex]\[
= -45 + 288 - 243
\][/tex]
[tex]\[
= 0
\][/tex]
So the determinant of the numerator matrix is 0.
Next, we find the determinant of the denominator matrix:
[tex]\[
\left| \begin{array}{rrr} -3 & 2 & -3 \\ 4 & -3 & 6 \\ 9 & -6 & 9 \end{array} \right|
\][/tex]
Using the same formula:
[tex]\[
= -3 \left( (-3)(9) - (6)(-6) \right) - 2 \left( 4(9) - (6)(9) \right) + (-3) \left( 4(-6) - (-3)(9) \right)
\][/tex]
Calculate each term inside:
[tex]\[
-3 \left( -27 + 36 \right) - 2 \left( 36 - 54 \right) - 3 \left( -24 + 27 \right)
\][/tex]
[tex]\[
= -3 \left( 9 \right) - 2 \left( -18 \right) - 3 \left( 3 \right)
\][/tex]
[tex]\[
= -27 + 36 - 9
\][/tex]
[tex]\[
= 0
\][/tex]
So the determinant of the denominator matrix is 0 as well.
Since both determinants are 0, the expression we need to evaluate is:
[tex]\[
x = \frac{0}{0}
\][/tex]
This results in an undefined form. In mathematical terms, the value of [tex]\( x \)[/tex] is "not a number" (NaN), which means that [tex]\( x \)[/tex] cannot be determined from the given matrices.
