Define the matrix
𝑀
=
[
3

5

−
3

−
1

5

−
1

6

𝑥
−
3

0
]
.

Which of the following is
det
⁡
𝑀
?

Question

Define the matrix
𝑀
=
[
3
	
5
	
−
3

−
1
	
5
	
−
1

6
	
𝑥
−
3
	
0
]
.

Which of the following is 
det
⁡
𝑀
?

BlackTom AI · Accepted Answer

Determinant of a 3×3 can be computed by cofactor expansion or the Rule of Sarrus. Expanding along the third column is convenient because the (3,3) entry is 0.

1) Relevant knowledge points and concepts
- Determinant of a 3×3: det(A) = sum over j of a1j C1j (or expand along any row/column).
- Cofactor sign: Cij = (-1)^{i+j} times the minor (determinant of the 2×2 submatrix after removing row i and column j).
- Choosing a row/column with a zero reduces work.

Matrix M:
[3   5   −3
 −1  5   −1
 6  x−3  0]

2) Complete reasoning process (expansion along column 3)
det(M) = a13·C13 + a23·C23 + a33·C33.
Here a13 = −3, a23 = −1, a33 = 0.

3) Calculations with intermediate steps
- For entry (1,3): sign = (-1)^{1+3} = +1. Minor M13 is determinant of
  [−1  5
   6  x−3]
  = (−1)(x−3) − (5·6)
  = −(x−3) − 30
  = −x + 3 − 30
  = −x − 27.
  Contribution from a13: (−3) · (+1) · (−x − 27) = −3(−x − 27) = 3x + 81.

- For entry (2,3): sign = (-1)^{2+3} = −1. Minor M23 is determinant of
  [3  5
   6  x−3]
  = 3(x−3) − 5·6
  = 3x − 9 − 30
  = 3x − 39.
  Cofactor C23 = −1·(3x − 39) = −3x + 39.
  Contribution from a23: (−1) · (−3x + 39) = 3x − 39.

- For entry (3,3): a33 = 0 so contribution = 0.

Add contributions: (3x + 81) + (3x − 39) = 6x + 42.

4) Why the correct answer is correct
The algebraic combination of cofactors and minors yields 6x + 42; signs and arithmetic were handled per cofactor rules, so the result is exact.

5) Why other options are incorrect
No alternative choices are provided here; common incorrect results would arise from sign errors in cofactors or arithmetic mistakes (for example forgetting the −1 factor for C23 or mis-evaluating 3(x−3) − 30).

6) Conclusion
The determinant det(M) = 6x + 42, which matches the provided correct answer.

Define the matrix 𝑀 = [ 3 5 − 3 − 1 5 − 1 6 𝑥 − 3 0 ] . Which of the following is det ⁡ 𝑀 ?单项选择题

类似问题

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