Home
Class 12
MATHS
Show that the matrix B^T\ A B is symmetr...

Show that the matrix `B^T\ A B` is symmetric or skew-symmetric according as `A` is symmetric or skew-symmetric.

Text Solution

Verified by Experts

If `A` be symmetric i.e., `A^{T}=A` then
`(B^{T} AB)^{T}=[B^{T}(AB)]^{T}=(AB)^{T}(B^{T})^{T} `
`=(B^{T} A^{T}) B=B^{T} A^{T} B=B^{T} AB`
Hence `B^{T} AB` is symmetric
If `A` is skew symmetric i.e., `A^{T}=-A` then
`(B^{T} A B)^{T}=[B^{T}(A B)]^{T}=(A B)^{T}(B^{T})^{T} `
...
Doubtnut Promotions Banner Mobile Dark
|

Topper's Solved these Questions

  • ADJOINTS AND INVERSE OF MATRIX

    RD SHARMA|Exercise QUESTION|1 Videos

  • ALGEBRA OF VECTORS

    RD SHARMA|Exercise Solved Examples And Exercises|331 Videos

Similar Questions

Explore conceptually related problems

Show that the matrix B^(T)AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.

Show that the matrix B^(^^), AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Knowledge Check

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

    A
    diagonal matrix
    B
    null matrix
    C
    triangular materix
    D
    none of these

    • Similar Questions

    Explore conceptually related problems

    If B is a square matrix and A is any square matrix of order equl to that of B, prove that B AB is symmetirc or skew symmetric according as A is symmetric or skew symmetric.

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

    if B is skew symmetric matrix then find matrix ABA^(T) is symmetric or skew symmetric

    If A is a symmetric matrix,write whether A^(T) is symmetric or skew-symmetric.

    For any square matrix write whether AA^(T) is symmetric or skew-symmetric.

    If B is a symmetric matrix,write whether the matrix ABA^(T) is symmetric or skew- symmetric.

    Let Aa n dB be two nonsinular square matrices, A^T a n dB^T are the transpose matrices of Aa n dB , respectively, then which of the following are correct? B^T A B is symmetric matrix if A is symmetric B^T A B is symmetric matrix if B is symmetric B^T A B is skew-symmetric matrix for every matrix A B^T A B is skew-symmetric matrix if A is skew-symmetric

    Home
    Profile