Matrices & Determinants
Equality of Matrices
Grade 12
Question:
<p>If <span style='display: inline-block; border: 1px solid black; padding: 5px;'><span style='display: inline-block; padding: 2px 5px;'>x + 3</span> <span style='display: inline-block; padding: 2px 5px;'>2y + x</span> <br/> <span style='display: inline-block; padding: 2px 5px;'>z - 1</span> <span style='display: inline-block; padding: 2px 5px;'>4w - 8</span></span> = <span style='display: inline-block; border: 1px solid black; padding: 5px;'><span style='display: inline-block; padding: 2px 5px;'>-x - 1</span> <span style='display: inline-block; padding: 2px 5px;'>0</span> <br/> <span style='display: inline-block; padding: 2px 5px;'>3</span> <span style='display: inline-block; padding: 2px 5px;'>2w</span></span>, then find the value of \(x + y + z + w\).</p>
Step-by-Step Solution
Key Concept: Two matrices are equal if and only if their corresponding elements are equal. Use this property to set up equations for each position.
<p><strong>Step 1:</strong> As the given matrices are equal, their corresponding elements are equal.</p><p><strong>Step 2:</strong> From the first element: $x + 3 = -x - 1 \Rightarrow 2x = -4 \Rightarrow x = -2$</p><p><strong>Step 3:</strong> From the second element: $2y + x = 0 \Rightarrow 2y - 2 = 0 \Rightarrow y = 1$</p><p><strong>Step 4:</strong> From the third element: $z - 1 = 3 \Rightarrow z = 4$</p><p><strong>Step 5:</strong> From the fourth element: $4w - 8 = 2w \Rightarrow 2w = 8 \Rightarrow w = 4$</p><p><strong>Step 6:</strong> Therefore, $x + y + z + w = -2 + 1 + 4 + 4 = 9$</p>
Correct Answer: 9