Question:
Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................
Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................
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For triangular matrices, determinants equal the product of diagonal elements — a key simplification in such problems.
Updated On: Dec 3, 2025
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Correct Answer: -216
Solution and Explanation
Step 1: Recognize that determinant of a power is the power of the determinant.
\[
\det((8I - M)^3) = (\det(8I - M))^3.
\]
Step 2: Note that \( M \) is upper triangular.
So, \( \det(8I - M) = \prod_{i=1}^4 (8 - m_{ii}) = (8 - 9)(8 - 7)(8 - 11)(8 - 0). \)
Step 3: Simplify.
\[
\det(8I - M) = (-1)(1)(-3)(8) = (-1 \times 1 \times -3 \times 8) = 24.
\]
Then,
\[
\det((8I - M)^3) = (24)^3 = 13824.
\]
Final Answer: \[ \boxed{13824} \]
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