AP EAMCET202124 Aug 2021Morning ShiftMathematicsMatricesActual
If a square matrix A is such that (A^T- 1 2 I ) (A- 1 2 I )= (A^T+ 1 2 I ) (A+ 1 2 I )=I , where I is a unit matrix, then A is
Options
- Asymmetric matrix
- Bequal to 3 4 ।
- Cskew-symmetric matrix
- Dequal to -3 4 ।
Correct answer
C. skew-symmetric matrix
Step-by-step solution
aligned Given, & (A^T- 1 2 I ) (A- 1 2 I ) = & (A^T+ 1 2 I ) (A+ 1 2 I )=I aligned Taking starting two, we get aligned & A^T A- 1 2 A^T I- 1 2 I A+ 1 4 I^2=A^T A+ 1 2 A^T I+ 1 2 I A & + 1 4 I^2 & - 1 2 A^T- 1 2 A= 1 2 A^T+ 1 2 A aligned [U sing cancellation law) and I A=I ] aligned & & 0 & = 1 2 A^ T + 1 2 A^T+ 1 2 A+ 1 2 A & & & =A^T+A & & A^T & =-A aligned A is skew-symmetric matrix.