Most Important Selected Qs for JEE AdvancedMathematicsMatrices
Which of the following is(are) correct?
Options
- AIf A and B are two square matrices of order 3 and A is a non-singular matrix such that A B=O , then B must be
- BIf A, B, C are three square matrices of order 2 and det. (A)=2 , det. (B)=3 , det. (C)=4 , then the value of d
- CIf A is a square matrix of order 3 and det. (A)= 1 2 , then det ( adj . A⁻¹ ) is 8 .
- DEvery skew symmetric matrix is singular.
Correct answer
B. If A, B, C are three square matrices of order 2 and det. (A)=2 , det. (B)=3 , det. (C)=4 , then the value of d
Step-by-step solution
(1) Given that A B=O , where det (A) 0 ...(1) So, A⁻¹ exists. Now, pre-multiplying equation (1) with A⁻¹ , we get (A⁻¹ A ) B=A⁻¹ O B=O_ rull matrix (2) Given, det. (A)=2 , det. (B)=3 , det. (C)=4 So, det. (3 A B C)=3^2 det. (A) det . (B) det. (C)=9(2)(3)(4)=216 (As, A, B, C are square matrices of order 2.) (3) Given, det. (A)= 1 2 (order of matrix A is 3 ) As, det. ( adj A)=( det . A)^ n-1 ...(1) place A by A⁻¹ in equation (1) and take n=3 , we get det ( adj A⁻¹ )= |A⁻¹ |^2= 1 |A|^2 = 1 ( 1 2 )^2 =4 (4) We know tha