<p><em>A</em> is an involuntary matrix given by \(A = \begin{bmatrix} 0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}\), then the inverse of \(A/2\) will be</p>
Step-by-Step Solution
Key Concept: For an involuntory matrix A where A² = I, we have A⁻¹ = A. For the matrix A/2, we need (A/2)⁻¹ = 2A⁻¹ = 2A using the scalar property of inverse matrices.
<p><strong>Step 1:</strong> Recognize that A is involuntory, meaning A² = I, which implies A⁻¹ = A.</p><p><strong>Step 2:</strong> Apply the scalar property of matrix inverses: for any non-zero scalar k, (kA)⁻¹ = (1/k)A⁻¹.</p><p><strong>Step 3:</strong> Calculate (A/2)⁻¹ = (1/(1/2)) × A⁻¹ = 2A⁻¹.</p><p><strong>Step 4:</strong> Since A⁻¹ = A (from the involuntory property), we get (A/2)⁻¹ = 2A.</p><p>∴ Answer: A (which represents the matrix 2A)</p>
Correct Answer: A