Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
<p>The determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mi>x</mi></mtd><mtd><mi>z</mi><mo>+</mo><mi>x</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>z</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>x</mi></mtd><mtd><mi>x</mi><mo>+</mo><mi>y</mi></mtd></mtr></mtable></mfenced></math> equals -</p>
(A) x<sup>2</sup>y<sup>2</sup>z<sup>2</sup>
(B) 4x<sup>2</sup>y<sup>2</sup>z<sup>2</sup>
(C) xyz
(D) 4xyz
Step-by-Step Solution
Key Concept: Use row and column operations to simplify the determinant. Specifically, perform R1 -> R1 - R3 and then expand or simplify further.
<p>Let Δ = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>x</mi></mtd><mtd><mi>x</mi></mtd><mtd><mi>z</mi><mo>+</mo><mi>x</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>z</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>x</mi></mtd><mtd><mi>x</mi><mo>+</mo><mi>y</mi></mtd></mtr></mtable></mfenced></math>. Applying R1 → R1 - R3, we get Δ = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>x</mi><mo>-</mo><mi>z</mi></mtd><mtd><mi>x</mi><mo>-</mo><mi>x</mi></mtd><mtd><mi>z</mi><mo>+</mo><mi>x</mi><mo>-</mo><mi>x</mi><mo>-</mo><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>z</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>x</mi></mtd><mtd><mi>x</mi><mo>+</mo><mi>y</mi></mtd></mtr></mtable></mfenced></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>x</mi><mo>-</mo><mi>z</mi></mtd><mtd><mn>0</mn></mtd><mtd><mi>z</mi><mo>-</mo><mi>y</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>y</mi></mtd><mtd><mi>z</mi></mtd></mtr><mtr><mtd><mi>z</mi></mtd><mtd><mi>x</mi></mtd><mtd><mi>x</mi><mo>+</mo><mi>y</mi></mtd></mtr></mtable></mfenced></math>. Expanding this determinant yields 4xyz.</p>
Correct Answer: 4