If \( A = \begin{bmatrix} K & 4 \\ 4 & K \end{bmatrix} \) and \( |A^3| = 729 \), then the value of \( K^8 \) is:
- \( 9^8 \)
\( 5^8 \)
\( 3^8 \)
- \( (-3)^8 \)
The Correct Option is B
Solution and Explanation
The determinant of A is:
$|A| = \begin{vmatrix} K & 4 \\ 4 & K \end{vmatrix} = K \cdot K - 4 \cdot 4 = K^2 - 16.$
Using the property of determinants:
$|A|^3 = (|A|)^3 = 729.$
Take the cube root:
$|A| = \sqrt[3]{729} = 9.$
Thus:
$K^2 - 16 = 9 \implies K^2 = 25.$
Therefore:
$K = \pm 5.$
The value of $K^8$ is:
$K^8 = (K^2)^4 = 25^4.$
Calculate:
$25^4 = (25^2)^2 = 625^2 = 390625.$
Final Answer:
$5^8$
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