Question: Compute the determinant using a cofactor expansion across the first row. Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
A.
Using this expansion, the determinant is
(negative 3)(negative 5) minus (1)(negative 6) plus (4)(negative 8) equals enter your response here.
B.
Using this expansion, the determinant is minus (negative 3)(negative 5) plus (1)(negative 6) minus (4)(negative 8) equals enter your response here.
C.
Using this expansion, the determinant is (2)(negative 9) minus (negative 3)(negative 5) plus (4)(11) equals enter your response here.
D.
Using this expansion, the determinant is minus (2)(negative 9) plus (negative 3)(negative 5) minus (4)(11) equals enter your response here.
1. **Problem Statement:** Compute the determinant of a matrix using cofactor expansion across the first row. Four options (A, B, C, D) are given with different expansions.
2. **Recall the cofactor expansion formula:** For a 3x3 matrix with first row elements $a_{11}, a_{12}, a_{13}$ and their cofactors $C_{11}, C_{12}, C_{13}$, the determinant is:
$$\det = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$$
Note that cofactors include signs: $C_{ij} = (-1)^{i+j} M_{ij}$ where $M_{ij}$ is the minor.
3. **Evaluate each option:**
**Option A:**
$$(-3)(-5) - (1)(-6) + (4)(-8)$$
Calculate step-by-step:
$$(-3)(-5) = 15$$
$$ (1)(-6) = -6$$
$$ (4)(-8) = -32$$
So,
$$15 - (-6) + (-32) = 15 + 6 - 32 = (21 - 32) = -11$$
**Option B:**
$$-(-3)(-5) + (1)(-6) - (4)(-8)$$
Calculate step-by-step:
$$-(-3)(-5) = - (15) = -15$$
$$ (1)(-6) = -6$$
$$ -(4)(-8) = +32$$
So,
$$-15 - 6 + 32 = 11$$
**Option C:**
$$(2)(-9) - (-3)(-5) + (4)(11)$$
Calculate step-by-step:
$$(2)(-9) = -18$$
$$-(-3)(-5) = -15$$
$$(4)(11) = 44$$
So,
$$-18 - 15 + 44 = 11$$
**Option D:**
$$-(2)(-9) + (-3)(-5) - (4)(11)$$
Calculate step-by-step:
$$-(2)(-9) = 18$$
$$(-3)(-5) = 15$$
$$-(4)(11) = -44$$
So,
$$18 + 15 - 44 = -11$$
4. **Summary of results:**
- Option A: $-11$
- Option B: $11$
- Option C: $11$
- Option D: $-11$
5. **Conclusion:** The correct choice depends on the original matrix and sign pattern. Since the problem asks to select the correct choice and fill in the answer, and options B and C both equal $11$, while A and D equal $-11$, the correct answer is either B or C depending on the matrix.
Since the problem states "using a cofactor expansion across the first row," and option C matches the standard sign pattern (+, -, +) for cofactors in the first row, option C is the correct expansion.
**Final answer:**
$$\boxed{11}$$