Consider the 3 × 3 matrix \(M = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 &…
2024
Consider the 3 × 3 matrix \(M = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 1 & 3 \\ 4 & 3 & 6 \end{bmatrix} \).
The determinant of (𝑴𝟐 + 12𝑴) is ______.
Attempted by 9 students.
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Correct answer: 0
Key idea: factor M^2 + 12M as M(M + 12I), so det(M^2 + 12M) = det(M) · det(M + 12I).
Step 1: Compute det(M). Using expansion along the first row:
det(M) = 1·det([[1,3],[3,6]]) - 2·det([[3,3],[4,6]]) + 3·det([[3,1],[4,3]])
= 1·(1·6 - 3·3) - 2·(3·6 - 3·4) + 3·(3·3 - 1·4)
= 1·(-3) - 2·6 + 3·5 = -3 - 12 + 15 = 0.
Step 2: Since det(M) = 0, the product det(M) · det(M + 12I) = 0. Therefore det(M^2 + 12M) = 0.
Answer: 0