Matrices & Determinants
Properties of Adjoint and Determinants
Grade 12
Question:
<p>Let <em>A</em> be an <em>n</em>th-order square matrix and <em>B</em> be its adjoint, then \(|AB + KI_n|\) is (where <em>K</em> is a scalar quantity)</p>
<p>\((|A| + K)^{n-2}\)</p>
<p>\((|A| + K)^n\)</p>
<p>\((|A| + K)^{n-1}\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: The product of a matrix with its adjoint gives A·adj(A) = |A|·I_n, so AB = |A|·I_n. This transforms the determinant expression into |A|·I_n + KI_n| = |(|A| + K)I_n|.
<p><strong>Step 1:</strong> Recall the fundamental property of adjoint matrices: <br/>A · adj(A) = |A| · I_n</p><p><strong>Step 2:</strong> Since B = adj(A), we have:<br/>AB = A · adj(A) = |A| · I_n</p><p><strong>Step 3:</strong> Substitute into the determinant expression:<br/>|AB + KI_n| = ||A|·I_n + KI_n| = |(|A| + K)I_n|</p><p><strong>Step 4:</strong> For a scalar multiple of the identity matrix:<br/>|(|A| + K)I_n| = (|A| + K)^n</p><p>∴ Answer: (|A| + K)^n or equivalent form (typically option B)</p>
Correct Answer: B