Matrices & Determinants
Inverse of a matrix
Grade Class 12
Question:
If A<sup>-1</sup> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math>, then
(A) |A| = 2
(B) A is non-singular
(C) Adj.A = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1/2</mn></mtd><mtd><mo>-</mo><mn>1/2</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>1/2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1/2</mn></mtd></mtr></mtable></mfenced></math>
(D) A is skew symmetric matrix
Step-by-Step Solution
Key Concept: Use the properties of inverse matrices: A^-1 = (Adj A) / |A|. Calculate |A^-1| to find |A| = 1/|A^-1|. Then find Adj A = |A| * A^-1.
Given A<sup>-1</sup> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math>. The determinant |A<sup>-1</sup>| = 1(2 - 0) - (-1)(0 - 0) + 0 = 2. Since |A<sup>-1</sup>| = 1/|A|, we have |A| = 1/2. Thus, A is non-singular. Adj A = |A| * A<sup>-1</sup> = (1/2) * <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1/2</mn></mtd><mtd><mo>-</mo><mn>1/2</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>1/2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1/2</mn></mtd></mtr></mtable></mfenced></math>. Options (A) and (B) are incorrect based on |A|=1/2, but the provided answer key for this question in the source is 'A', which seems to be a typo in the source document's answer key for this specific question number. Based on standard mathematical derivation, (B) and (C) are correct.
Correct Answer: 1, 2