MHT CET202520 Apr 2025Morning ShiftMathematicsMatricesActual
If A = [ array cc 1 & x - x & 1 array ] , then A ^ T A ⁻¹=
Options
- A[ array cc 2 x & - 2 x - 2 x & 2 x array ]
- B[ array lr 2 x & - 2 x 2 x & 2 x array ]
- C[ array cc - 2 x & 2 x 2 x & 2 x array ]
- D[ array ll - 2 x & 2 x - 2 x & 2 x array ]
Correct answer
B. [ array lr 2 x & - 2 x 2 x & 2 x array ]
Step-by-step solution
The transpose of A = bmatrix 1 & x - x & 1 bmatrix is A^T = bmatrix 1 & - x x & 1 bmatrix . The determinant evaluates to |A| = (1)(1) - ( x)(- x) = 1 + ^2 x = ^2 x , using the trigonometric identity. The adjugate matrix is bmatrix 1 & - x x & 1 bmatrix , leading to the inverse: A⁻¹ = 1 ^2 x bmatrix 1 & - x x & 1 bmatrix = bmatrix ^2 x & - x x x x & ^2 x bmatrix Multiplying matrices: A^T A⁻¹ = bmatrix 1 & - x x & 1 bmatrix bmatrix ^2 x & - x x x x & ^2 x bmatrix The product yields bmatrix ^2 x - ^2 x & -2 x x 2 x x