Define the matrix
M=[4 1 −4
−2 x−4 −1
0 3 −1].

Which of the following is detM?

Question

Define the matrix
M=[4	1	−4
−2	x−4	−1
0	3	−1].

Which of the following is detM?

BlackTom AI · Accepted Answer

The problem asks for det M where M is the 3x3 matrix given with x in the (2,2) position.
Option checks: there are no listed answer choices provided in the prompt, so I will derive det M step by step so you can see the result that a student would compare against any potential option.
I begin by expanding along the first row, since it often keeps the arithmetic organized.
Computing the first minor associated with the entry in row 1, column 1 (which is 4): we take the determinant of the 2x2 matrix formed by deleting row 1 and column 1, namely [[x-4, -1], [3, -1]]. Its determinant is (x-4)*(-1) - (-1)*3 = -(x-4) + 3 = -x + 7. Multiplying by the cofactor 4 gives 4*(-x + 7) = -4x + 28.
Next, for the entry in row 1, column 2 (which is 1) we remove row 1 and column 2 to get the 2x2 matrix [[-2, -1], [0, -1]]. Its determinant is (-2)*(-1) - (-1)*0 = 2 - 0 = 2. The cofactor is (-1)^{1+2} times this determinant, i.e., -1 * 2 = -2, and multiplying by the original entry (which is 1) yields -2.
Finally, for the entry in row 1, column 3 (which is -4) we delete row 1 and column 3 to form [[-2, x-4], [0, 3]]. The determinant of this 2x2 is (-2)*3 - (x-4)*0 = -6 - 0 = -6. The cofactor is (-1)^{1+3} times this determinant, i.e., (+1)*(-6) = -6, and multiplying by the original entry (-4) gives (-4)*(-6) = 24.
Now sum the contributions from the three terms in the first row: (-4x + 28) + (-2) + 24 = -4x + 50.
Therefore, det M = 50 - 4x.
If an answer option were present, it would need to match this expression, 50 − 4x. Since no answer choices were provided in the prompt, the computed determinant as a function of x is 50 − 4x.
Notes on potential pitfalls: be careful with the signs during cofactor expansion, especially the alternating +/- pattern in the first row; each minor determinant must be computed correctly before applying the sign. A common error is miscalculating one 2x2 determinant or misapplying the sign for the cofactor, which would lead to an incorrect final expression.
If you want, I can re-derive using a different expansion method (e.g., expanding along a different row or column) to show the same result for verification.

MATH1061/1002/1021 (ND) MATH1061 Canvas Quiz 9

Define the matrix M=[4 1 −4 −2 x−4 −1 0 3 −1]. Which of the following is detM?

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