Home
Class 12
MATHS
If A is symmetric and B skew- symmetric ...

If A is symmetric and B skew- symmetric matrix and `A + B ` is
non-singular and `C= (A+B) ^(-1) (A-B)`
`C^(T) AC` equals to

A

`A + B`

B

`A-B`

C

A

D

B

Text Solution

Verified by Experts

Given, `A^(T) = A, B^(T) = -B, det (A+ B) ne 0`
and `C = (A + B)^(-1) (A-B)`
`rArr (A + B) C = A - B` ...(i)
Also, ` (A + B) ^(T)= A - B` ...(ii)
and ` (A - B) ^(T)= A + B` ...(iii)
`C^(T) (A-B) = [C^(T) (A+B) ^(T)]C` [from Eq. (ii)]
`= [(A+ B)C]^(T) C`
`= (A - B) C` [from Eq. (i)]
`=(A + B) C` [from Eq. (iii) ]
` = A - B ` [ffrom Eq. (i)]
Promotional Banner

Similar Questions

Explore conceptually related problems

Define a skew-symmetric matrix.

If A is symmetric as well as skew-symmetric matrix, then A is

If A is a symmetric matrix, B is a skew-symmetric matrix, A+B is nonsingular and C=(A+B)^(-1) (A-B) , then prove that (i) C^(T) (A+B) C=A+B (ii) C^(T) (A-B)C=A-B (iii) C^(T)AC=A

If A is a symmetric matrix, B is a skew-symmetric matrix, A+B is nonsingular and C=(A+B)^(-1) (A-B) , then prove that (i) C^(T) (A+B) C=A+B (ii) C^(T) (A-B)C=A-B (iii) C^(T)AC=A

If A and B are symmetric matrices, then A B A is

If a matrix A is both symmetric and skew symmetric, then

If the matrix A is both symmetric and skew symmetric, then

If A is a skew-symmetric matrix and n is odd positive integer, then A^n is