Matrices & Determinants
Properties of determinants
Grade Class 12
Question:
For positive numbers x, y and z, the numerical value of the determinant <br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><msub><mi>log</mi><mi>x</mi></msub><mi>y</mi></mtd><mtd><msub><mi>log</mi><mi>x</mi></msub><mi>z</mi></mtd></mtr><mtr><mtd><msub><mi>log</mi><mi>y</mi></msub><mi>x</mi></mtd><mtd><mn>1</mn></mtd><mtd><msub><mi>log</mi><mi>y</mi></msub><mi>z</mi></mtd></mtr><mtr><mtd><msub><mi>log</mi><mi>z</mi></msub><mi>x</mi></mtd><mtd><msub><mi>log</mi><mi>z</mi></msub><mi>y</mi></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <br> is -
(A) 0
(B) log xyz
(C) log(x + y + z)
(D) logx logy logz
Step-by-Step Solution
Key Concept: Use the property of logarithms log_a b = (log b) / (log a). The determinant becomes a product of a matrix and its transpose or can be shown to have linearly dependent rows/columns.
Let the determinant be D. Using the property log_a b = (ln b) / (ln a), the determinant can be written as: <br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mfenced open="|"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mfrac><mrow><mi>ln</mi><mo> </mo><mi>y</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>ln</mi><mo> </mo><mi>z</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>y</mi></mrow></mfrac></mtd><mtd><mn>1</mn></mtd><mtd><mfrac><mrow><mi>ln</mi><mo> </mo><mi>z</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>y</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>z</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>ln</mi><mo> </mo><mi>y</mi></mrow><mrow><mi>ln</mi><mo> </mo><mi>z</mi></mrow></mfrac></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <br> Taking 1/(ln x), 1/(ln y), and 1/(ln z) common from R1, R2, and R3 respectively: <br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>y</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>z</mi></mrow></mfrac><mfenced open="|"><mtable><mtr><mtd><mi>ln</mi><mo> </mo><mi>x</mi></mtd><mtd><mi>ln</mi><mo> </mo><mi>y</mi></mtd><mtd><mi>ln</mi><mo> </mo><mi>z</mi></mtd></mtr><mtr><mtd><mi>ln</mi><mo> </mo><mi>x</mi></mtd><mtd><mi>ln</mi><mo> </mo><mi>y</mi></mtd><mtd><mi>ln</mi><mo> </mo><mi>z</mi></mtd></mtr><mtr><mtd><mi>ln</mi><mo> </mo><mi>x</mi></mtd><mtd><mi>ln</mi><mo> </mo><mi>y</mi></mtd><mtd><mi>ln</mi><mo> </mo><mi>z</mi></mtd></mtr></mtable></mfenced></math> <br> Since all rows are identical, the value of the determinant is 0.
Correct Answer: 1