Matrices & Determinants
Equality of matrices
Grade Class 12
Question:
If <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mn>3</mn><mi>x</mi><mo>-</mo><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi><mi>x</mi><mo>+</mo><mi>z</mi></mtd><mtd><mn>3</mn><mi>y</mi><mo>-</mo><mi>w</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></math>, then
(A) x = 3, y = 7, z = 1, w = 14
(B) x = 3, y = -5, x = -1, w = -4
(C) x = 3, y = 6, z = 2, w = 7
(D) None of these
Step-by-Step Solution
Key Concept: Equating corresponding elements of two equal matrices.
Given <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mn>3</mn><mi>x</mi><mo>-</mo><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi><mi>x</mi><mo>+</mo><mi>z</mi></mtd><mtd><mn>3</mn><mi>y</mi><mo>-</mo><mi>w</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></math>. Equating elements: x = 3. 3x - y = 2 => 3(3) - y = 2 => 9 - y = 2 => y = 7. zx + z = 4 => z(x + 1) = 4 => z(3 + 1) = 4 => 4z = 4 => z = 1. 3y - w = 7 => 3(7) - w = 7 => 21 - w = 7 => w = 14. Thus, x = 3, y = 7, z = 1, w = 14.
Correct Answer: 1