MHT CET202523 Apr 2025Morning ShiftMathematicsMatricesActual
If A and B are non-singular matrices of order 2 such that ( AB )⁻¹= 1 6 [ array cc -7 & -3 2 & 3 array ] and A ⁻¹= 1 3 [ array cc 4 & 3 -1 & 0 array ] then B ⁻¹=
Options
- A1 2 [ array cc 2 & 3 1 & -1 array ]
- B1 2 [ array ll 3 & 1 2 & 4 array ]
- C1 2 [ array cc -1 & 3 1 & 2 array ]
- D1 6 [ array ll 1 & 1 2 & 3 array ]
Correct answer
C. 1 2 [ array cc -1 & 3 1 & 2 array ]
Step-by-step solution
Inverse matrix computation: The inverse of the product AB relates to the inverses of A and B by (AB)⁻¹ = B⁻¹A⁻¹ . Given (AB)⁻¹ = 1 6 bmatrix -7 & -3 2 & 3 bmatrix and A⁻¹ = 1 3 bmatrix 4 & 3 -1 & 0 bmatrix . Multiply both sides of (AB)⁻¹ = B⁻¹A⁻¹ on the right by A : B⁻¹ = (AB)⁻¹A = ( 1 6 bmatrix -7 & -3 2 & 3 bmatrix )A To determine A , invert A⁻¹ . Let C = bmatrix 4 & 3 -1 & 0 bmatrix , so A⁻¹ = 1 3 C and A = 3C⁻¹ . The determinant of C is (C) = 4 0 - 3 (-1) = 3 . The adjugate is adj (C) = bmatrix 0 & -3 1 & 4 bma