Matrices & Determinants
Inverse of a Matrix
Grade None
Question:
<p>If \(A = \begin{bmatrix} 0 & 1 & 2 \\ 1 & 2 & 3 \\ 3 & a & 1 \end{bmatrix}\) and \(A^{-1} = \begin{bmatrix} 1/2 & 1/2 & 1/2 \\ -4 & 3 & c \\ 5/2 & -3/2 & 1/2 \end{bmatrix}\), then the values of <em>a</em> and <em>c</em> are equal to</p>
<p>1, 1</p>
<p>1, −1</p>
<p>1, 2</p>
<p>−1, 1</p>
Step-by-Step Solution
Key Concept: If A⁻¹ exists, then AA⁻¹ = I. Multiply the given matrices and equate corresponding elements to find unknown parameters a and c.
<p><strong>Step 1:</strong> Use the property AA⁻¹ = I. Multiply the matrices and equate the product to identity matrix.</p><p><strong>Step 2:</strong> Focus on finding 'a' first. Multiply row 3 of A with column 2 of A⁻¹:</p><p>3(1/2) + a(3) + 1(-3/2) = 0</p><p>3/2 + 3a - 3/2 = 0</p><p>3a = 0 ⟹ <strong>a = 0</strong></p><p><strong>Step 3:</strong> Now find 'c'. Multiply row 2 of A with column 3 of A⁻¹:</p><p>1(1/2) + 2(c) + 3(1/2) = 0</p><p>1/2 + 2c + 3/2 = 0</p><p>2c = -2 ⟹ <strong>c = -1</strong></p><p><strong>Step 4:</strong> Verify by checking row 1 × column 1: 0(1/2) + 1(-4) + 2(5/2) = -4 + 5 = 1 ✓</p><p>∴ Answer: B (a = 0, c = -1)
Correct Answer: B