Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
If $A$ be $3 \times 3$ non-singular matrix, $|A| = K$, then $|(xA)^{-1}| = $ (where $x \neq 0$)
$xK$
$\frac{1}{xK}$
$\frac{1}{x^3 K^3}$
$\frac{1}{x^3 K}$
Step-by-Step Solution
Key Concept: For a scalar multiple of a matrix: |xA| = x^n|A| where n is the order of the matrix. Then apply the determinant property of inverse: |(xA)^{-1}| = 1/|xA|.
For the inverse of a matrix, $|(xA)^{-1}| = \frac{1}{|xA|} = \frac{1}{x^n|A|} = \frac{1}{x^n K}$ where $K$ is a scalar related to $|A|$.
Correct Answer: 4