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:

Question

For the matrices $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} $ and $ B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} $, if $ (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} $, then among the following which one is true?}

Answer 1

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 $m + 2n$.

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:

The Correct Option is C

Solution and Explanation

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