Question

If $A$ is a square matrix of order 3 such that \[ \det(A) = 3 \] and \[ \det(\text{adj}(-4 \, \text{adj}(-3 \, \text{adj}(3 \, \text{adj}((2A)^{-1}))))) = 2^{m^3 n}, \] then $m + 2n$ is equal to:

• A. 3
• B. 2
• C. 4
• D. 6

Hint:

When you see several nested adjugates and scalar multiples of the same matrix, avoid writing out full matrix expressions. Instead track only the powers of 2 and 3 in the determinant, using the rules for how adj and scalar multiplication affect the determinant of an n by n matrix, working from the innermost operation outward.

Answer 1

The Correct Option is C

Given \(|A| = 3\), the expression is:
\[\left| \text{adj} \left( -4 \, \text{adj} - 3 \, \text{adj} \left( 3 \, \text{adj} \left( 2A^{-1} \right) \right) \right) \right|\]
Step 1: Simplify the innermost expression.
Step 2: Expand the outer term.
Step 3: Replace the outermost adj with its expression.
Step 4: Simplify the term inside the absolute value.
Step 5: Use the property of adjugates.
Step 6: Replace \(\left| 2A^{-1} \right|^{16}\) with its determinant form.
Step 7: Substitute \(|2A|^{16} = 2^{16} \cdot |A|^{16}\).
Step 8: Replace \(|A| = 3\).
Step 9: Simplify powers of 2 and 3.
Step 10: Further simplify.
Step 11: Define \(m\) and \(n\).
Step 12: Calculate the final result of \(m + 2n\).