Question

Let \( A \) be a \(3 \times 3\) matrix and let \( B = 3A \). If \( |A| = 5 \), then the value of \( \frac{|\text{adj } B|}{|3A|} \) is equal to

• A. \(27 \)
• B. \(125 \)
• C. \(25 \)
• D. \(135 \)
• E. \(81 \)

Hint:

For adjoint: \( |\text{adj }A| = |A|^{n-1} \). For scalar multiple: \( |kA| = k^n |A| \).

Answer 1

The Correct Option is D

Note: The expression in the question, \(\frac{adj B}{3A}\), should be interpreted as \(\frac{|adj B|}{|3A|}\) because the options are scalar values.
Step 1: Understanding the Concept:
This problem requires the application of several properties of determinants and adjugate matrices, especially how they behave with scalar multiplication.
Step 2: Key Formula or Approach:
For an n \(\times\) n matrix M and a scalar k: 1. `|kM| = k^n |M|` 2. `|adj M| = |M|^(n-1)` We are given A is a 3x3 matrix, so n=3. We have `B = 3A` and `|A| = 5`. We need to compute \(\frac{|adj B|}{|3A|}\).
Step 3: Detailed Explanation:
1. Calculate the denominator, |3A|:
Using the property `|kM| = k^n |M|` with k=3 and n=3: \[ |3A| = 3^3 |A| = 27 \cdot |A| \] Since `|A|=5`, \[ |3A| = 27 \cdot 5 = 135 \] 2. Calculate the numerator, |adj B|:
First, we need `|B|`. Since `B=3A`, `|B| = |3A| = 135`. Now use the property `|adj M| = |M|^(n-1)` with M=B and n=3: \[ |adj B| = |B|^(3-1) = |B|^2 \] Substitute the value of `|B|`: \[ |adj B| = (135)^2 \] 3. Compute the final expression:
\[ \frac{|adj B|}{|3A|} = \frac{(135)^2}{135} \] \[ = 135 \] Step 4: Final Answer:
The value of the expression is 135.