Matrices & Determinants
Properties of Determinants
Grade Class 12
Question:
If the determinant <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi><mo>+</mo><mi>p</mi></mtd><mtd><mi>l</mi><mo>+</mo><mi>x</mi></mtd><mtd><mi>u</mi><mo>+</mo><mi>f</mi></mtd></mtr><mtr><mtd><mi>b</mi><mo>+</mo><mi>q</mi></mtd><mtd><mi>m</mi><mo>+</mo><mi>y</mi></mtd><mtd><mi>v</mi><mo>+</mo><mi>g</mi></mtd></mtr><mtr><mtd><mi>c</mi><mo>+</mo><mi>r</mi></mtd><mtd><mi>n</mi><mo>+</mo><mi>z</mi></mtd><mtd><mi>w</mi><mo>+</mo><mi>h</mi></mtd></mtr></mtable></mfenced></math> splits into exactly K determinants of order 3, each element of which contains only one term, then the value of K, is-
Step-by-Step Solution
Key Concept: The property of determinants states that if each element of a row (or column) is the sum of two terms, the determinant can be expressed as the sum of two determinants. Since each of the 3 rows has 2 terms, the total number of determinants formed is 2^3 = 8.
Each row of the determinant has 2 terms. Since there are 3 rows, each row can be split into 2 determinants. By the property of determinants, the total number of determinants formed is 2 * 2 * 2 = 2^3 = 8.
Correct Answer: (B)