Matrices & Determinants
Matrices And Determinants
nta_abhyas_2025
Grade 12
Question:
Let $d$ be a matrix of order $3 \times 3$ such that $\det(4) = 2$, $B = 2d$ and $C = \frac{1}{\sqrt[3]{4}}$, then the value of $\det\left(\frac{d^3(C^3)}{\sqrt[3]{d}}\right)$ is
Step-by-Step Solution
Key Concept: Properties of determinants include $|kA| = k^n|A|$ for $n \times n$ matrices, and $|adjA| = |A|^{n-1}$.
We compute $|A| = 2$, $|B| = |2A^{-1}| = 2^3 \cdot |A|^{-1} = 8 \cdot \frac{1}{2} = 4$, and $|C| = \left|\frac{adjA}{|AB|}\right| = \frac{|adjA|}{|AB|} = \frac{|A|^2}{|A||B|} = \frac{4}{8} = \frac{1}{2}$. Therefore, $\det(A^TB^2C^3) = |A| \cdot |B|^2 \cdot |C|^3 = 2 \cdot 16 \cdot \frac{1}{8} = 4$, giving the answer as option 2.
Correct Answer: 2