Define the matrix $$M = \left [\begin{matrix}1 & -5 & 1\3 & 0 & x + 6\2 & 3 & -5\end{matrix}\right ] .$$

Which of the following is $\operatorname {det}{M}$?

Question

Define the matrix $$M = \left [\begin{matrix}1 & -5 & 1\3 & 0 & x + 6\2 & 3 & -5\end{matrix}\right ] .$$

Which of the following is $\operatorname {det}{M}$?

BlackTom AI · Accepted Answer

The problem presents the matrix M = [ [1, -5, 1], [3, 0, x + 6], [2, 3, -5] ] and asks for det(M).
Since the answer options are missing, I will verify the determinant calculation and show how the given option -13x - 144 arises.
We compute det(M) using the first-row expansion:
- First term: 1 times the determinant of the submatrix [[0, x+6], [3, -5]] which is (0)(-5) - (x+6)(3) = -3(x+6) = -3x - 18.
- Second term: subtracting the middle element (-5) times the determinant of the submatrix [[3, x+6], [2, -5]]; but since the cofactor for the (1,2) position includes a negative sign, the contribution is +5 times det([[3, x+6], [2, -5]]). The determinant here is 3(-5) - (x+6)(2) = -15 - 2x - 12 = -2x - 27, so this term is 5(-2x - 27) = -10x - 135.
- Third term: + the (1,3) entry 1 times the determinant of the submatrix [[3, 0], [2, 3]] which is 3*3 - 0*2 = 9.
Summing these contributions: (-3x - 18) + (-10x - 135) + 9 = -13x - 144.
Thus det(M) = -13x - 144.
Given that the only listed option matches this expression, it is the correct choice among the provided options. Note: since no other options were supplied, there’s nothing to compare against beyond the single option shown, and the computation aligns with the standard determinant expansion.

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

Define the matrix \[M = \left [\begin{matrix}1 & -5 & 1\\3 & 0 & x + 6\\2 & 3 & -5\end{matrix}\right ] .\] Which of the following is \(\operatorname {det}{M}\)?

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