Answer :
To determine the correct determinant of matrix [tex]\( A \)[/tex], we need to follow these steps:
Given matrix [tex]\( A \)[/tex] is:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
First, we need to calculate the determinant of matrix [tex]\( A \)[/tex].
1. Construct the Full Matrix:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
2. Compute the Determinant:
For a 3x3 matrix [tex]\(A\)[/tex], the determinant is given by:
[tex]\[ \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \][/tex]
Applying this to our matrix:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
We need to calculate the determinant of the matrix inside, then multiply the result by [tex]\(\frac{1}{2}\)[/tex].
The determinant of
[tex]\[ \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
is calculated as:
[tex]\[ \text{det} = \left(\frac{k}{2}\right) \left(k \cdot 1 - (-1) \cdot 0 \right) - \left(\frac{9}{2}\right) \left(1 \cdot 1 - 0 \cdot 5 \right) + (1)(1 \cdot (-1) - 5 \cdot k) \][/tex]
Simplifying each term:
[tex]\[ = \left(\frac{k}{2} \cdot (k + 1)\right) - \left(\frac{9}{2} \cdot 1\right) + \left(1 \cdot (-1 - 5k)\right) \][/tex]
[tex]\[ = \left(\frac{k^2}{2} + \frac{k}{2}\right) - \frac{9}{2} + \left(-1 - 5k\right) \][/tex]
Combine the terms:
[tex]\[ = \frac{k^2}{2} + \frac{k}{2} - \frac{9}{2} - 1 - 5k \][/tex]
[tex]\[ = \frac{k^2}{2} + \frac{k}{2} - 5k - \frac{11}{2} \][/tex]
The determinant we need simplifying does:
[tex]\[ = \frac{k^2 - 10k - 11}{2} \][/tex]
Multiplying by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ = \frac{1}{2} \cdot \left(\frac{k^2 - 10k - 11}{2}\right) = \frac{k^2 - 10k - 11}{4} \][/tex]
3. Match the Simplified Determinant with the Options:
Simplify the options:
Option (a):
[tex]\[ \frac{1}{8}\left(-2k^2 - 3k + 5\right) \][/tex]
Option (b):
[tex]\[ \frac{1}{2}\left(-2k^2 - 9k + 11\right) \][/tex]
Option (c):
[tex]\[ \frac{1}{16}\left(k^2 - 10k - 11\right) = \frac{k^2 - 10k - 11}{16} \][/tex]
We see, Comparing [tex]\(\frac{k^2 - 10k - 11}{4} \)[/tex] with answers c's [tex]\(\frac{k^2 - 10k - 11}16\)[/tex]:
In conclusion, the correct answer is:
[tex]\[ (c) \; |A| = \frac{1}{16}\left(k^2 - 10k - 11\right) \][/tex]
Given matrix [tex]\( A \)[/tex] is:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
First, we need to calculate the determinant of matrix [tex]\( A \)[/tex].
1. Construct the Full Matrix:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
2. Compute the Determinant:
For a 3x3 matrix [tex]\(A\)[/tex], the determinant is given by:
[tex]\[ \text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg) \][/tex]
Applying this to our matrix:
[tex]\[ A = \frac{1}{2} \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
We need to calculate the determinant of the matrix inside, then multiply the result by [tex]\(\frac{1}{2}\)[/tex].
The determinant of
[tex]\[ \begin{pmatrix} \frac{k}{2} & \frac{9}{2} & 1 \\ 1 & k & 0 \\ 5 & -1 & 1 \end{pmatrix} \][/tex]
is calculated as:
[tex]\[ \text{det} = \left(\frac{k}{2}\right) \left(k \cdot 1 - (-1) \cdot 0 \right) - \left(\frac{9}{2}\right) \left(1 \cdot 1 - 0 \cdot 5 \right) + (1)(1 \cdot (-1) - 5 \cdot k) \][/tex]
Simplifying each term:
[tex]\[ = \left(\frac{k}{2} \cdot (k + 1)\right) - \left(\frac{9}{2} \cdot 1\right) + \left(1 \cdot (-1 - 5k)\right) \][/tex]
[tex]\[ = \left(\frac{k^2}{2} + \frac{k}{2}\right) - \frac{9}{2} + \left(-1 - 5k\right) \][/tex]
Combine the terms:
[tex]\[ = \frac{k^2}{2} + \frac{k}{2} - \frac{9}{2} - 1 - 5k \][/tex]
[tex]\[ = \frac{k^2}{2} + \frac{k}{2} - 5k - \frac{11}{2} \][/tex]
The determinant we need simplifying does:
[tex]\[ = \frac{k^2 - 10k - 11}{2} \][/tex]
Multiplying by [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ = \frac{1}{2} \cdot \left(\frac{k^2 - 10k - 11}{2}\right) = \frac{k^2 - 10k - 11}{4} \][/tex]
3. Match the Simplified Determinant with the Options:
Simplify the options:
Option (a):
[tex]\[ \frac{1}{8}\left(-2k^2 - 3k + 5\right) \][/tex]
Option (b):
[tex]\[ \frac{1}{2}\left(-2k^2 - 9k + 11\right) \][/tex]
Option (c):
[tex]\[ \frac{1}{16}\left(k^2 - 10k - 11\right) = \frac{k^2 - 10k - 11}{16} \][/tex]
We see, Comparing [tex]\(\frac{k^2 - 10k - 11}{4} \)[/tex] with answers c's [tex]\(\frac{k^2 - 10k - 11}16\)[/tex]:
In conclusion, the correct answer is:
[tex]\[ (c) \; |A| = \frac{1}{16}\left(k^2 - 10k - 11\right) \][/tex]
