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 determinant of a matrix where each column is a sum of two terms can be expanded using the linearity property of determinants. For a 3x3 determinant where each of the 3 columns is a sum of 2 terms, the total number of determinants formed is 2^3 = 8.
Each column of the given 3x3 determinant is a sum of two terms. By the property of determinants, if each column is a sum of two terms, the determinant can be expressed as the sum of 2^n determinants, where n is the order of the determinant. Here, n=3, so the number of determinants is 2^3 = 8.
Correct Answer: (B)