Home
Class 12
MATHS
If there are three square matrix A, B, C...

If there are three square matrix A, B, C of same order satisfying the equation `A^2=A^-1 and B=A^(2^n) and C=A^(2^((n-2))`, then prove that `det .(B-C) = 0, n in N`.

Text Solution

Verified by Experts

`because B = ^(2^(n)) = A^(2.2^(n-1)) = (A^(2))^(2^(n-1)) = (A^(-1)) ^(2^(n-1) ) [ because A^(2) = A^(-1)]`
`= (A^(2^(n-1)))^(-1) = (A^(2.2^(n-2)))^(-1) = [(A^(2))^(2^(n-2))]^(-1)`
`= [(A^(-1))^(2^(n-2))]^(-1)= ((A^(-1))^(-1))^(2^(n-2)) = A^(2^(n-2))=C`
`rArr B-C=0 rArr det (B-C) = 0`
Promotional Banner

Similar Questions

Explore conceptually related problems

If A,B and C arae three non-singular square matrices of order 3 satisfying the equation A^(2)=A^(-1) let B=A^(8) and C=A^(2) ,find the value of det (B-C)

Let A be a square matrix of order 3 satisfies the relation A^(3)-6A^(2)+7A-8I=O and B=A-2I . Also, det. A=8 , then

For a square matrix A and a non singular matrix B of the same order , the value of det (B^(-1)AB) is :

If a square matrix A is involutory, then A^(2n+1) is equal to:

If the roots of the equations (b-c) x^(2) + (c-a) x+( a-b) =0 are equal , then prove that 2b=a+c

If A=[[a,b],[c,d]] (where b c!=0 ) satisfies the equations x^2+k=0,t h e n

The number of positive integers satisfying the inequality C(n+1,n-2) - C(n+1,n-1)<=100 is

The number of positive integers satisfying the inequality : ""^(n+1)C_(n-2)-""^(n+1)C_(n-1)le100 is :