<p>Find the values of \(a\), \(b\), \(c\), and \(d\) from the equation: \[\begin{bmatrix} a-b & 2a+c \\ 2a-b & 3c+d \end{bmatrix} = \begin{bmatrix} -1 & 5 \\ 0 & 13 \end{bmatrix}\]</p>
Step-by-Step Solution
Key Concept: Two matrices are equal if and only if their corresponding elements are equal. This creates a system of 4 linear equations in 4 unknowns that can be solved systematically by equating each position.
<p><strong>Step 1: Set up equations from matrix equality</strong></p><p>Equating corresponding elements:</p><p>Position (1,1): a - b = -1 ... (1)</p><p>Position (1,2): 2a + c = 5 ... (2)</p><p>Position (2,1): 2a - b = 0 ... (3)</p><p>Position (2,2): 3c + d = 13 ... (4)</p><p><strong>Step 2: Solve for a and b</strong></p><p>From equation (3): 2a - b = 0 ⟹ b = 2a</p><p>Substitute into equation (1): a - 2a = -1 ⟹ -a = -1 ⟹ <strong>a = 1</strong></p><p>Therefore: <strong>b = 2(1) = 2</strong></p><p><strong>Step 3: Solve for c</strong></p><p>From equation (2): 2(1) + c = 5 ⟹ c = 3</p><p>Therefore: <strong>c = 3</strong></p><p><strong>Step 4: Solve for d</strong></p><p>From equation (4): 3(3) + d = 13 ⟹ 9 + d = 13 ⟹ <strong>d = 4</strong></p><p><strong>Step 5: Verification</strong></p><p>Check all four equations: (1-2=-1)✓, (2+3=5)✓, (2-2=0)✓, (9+4=13)✓</p><p><strong>∴ Answer: a = 1, b = 2, c = 3, d = 4</strong></p>
Correct Answer: a=1, b=2, c=3, d=4