Advertisements
Advertisements
प्रश्न
Show that a matrix which is both symmetric and skew symmetric is a zero matrix.
Advertisements
उत्तर
Let A = [aij] be a matrix which is both symmetric and skew-symmetric.
Since A is a skew-symmetric matrix, so A′ = –A.
Thus for all i and j, we have aij = – aji ......(1)
Again, since A is a symmetric matrix, so A′ = A.
Thus, for all i and j, we have
aji = aij ......(2)
Therefore, from (1) and (2), we get
aij = – aij for all i and j
or
2aij = 0
i.e., aij = 0 for all i and j.
Hence A is a zero matrix.
APPEARS IN
संबंधित प्रश्न
If A = `[(-1, 2, 3),(5, 7, 9),(-2, 1, 1)]` and B = `[(-4, 1, -5),(1, 2, 0),(1, 3, 1)]`, then verify that (A – B)' = A' – B'
If A' = `[(3, 4),(-1, 2),(0, 1)]` and B = `[(-1, 2, 1),(1, 2, 3)]`, then verify that (A + B)' = A' + B'
If A' = `[(3, 4),(-1, 2),(0, 1)]` and B = `[(-1, 2, 1),(1, 2, 3)]`, then verify that (A – B)' = A' – B'
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
For the matrix A = `[(1, 5),(6, 7)]` verify that (A + A') is a symmetric matrix.
Find the values of x, y, z if the matrix A = `[(0, 2y, z),(x, y, -z),(x, -y, z)]` satisfy the equation A'A = I.
If the matrix A is both symmetric and skew symmetric, then ______.
Write a square matrix which is both symmetric as well as skew-symmetric.
If a matrix A is both symmetric and skew-symmetric, then
If A is a square matrix, then AA is a
If A and B are symmetric matrices, then ABA is
If A and B are matrices of the same order, then ABT − BAT is a
If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.
If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
Show that A′A and AA′ are both symmetric matrices for any matrix A.
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.
The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.
If A and B are matrices of same order, then (AB′ – BA′) is a ______.
If A is a skew-symmetric matrix, then A2 is a ______.
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A is symmetric matrix, then B′AB is ______.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
If A is skew-symmetric matrix, then A2 is a symmetric matrix.
If P is of order 2 x 3 and Q is of order 3 x 2, then PQ is of order ____________.
If A and B are symmetric matrices of the same order, then ____________.
If A is any square matrix, then which of the following is skew-symmetric?
The diagonal elements of a skew symmetric matrix are ____________.
If A, B are Symmetric matrices of same order, then AB – BA is a
If A = [aij] is a skew-symmetric matrix of order n, then ______.
For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?
Which of the following is correct?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its symmetric part \[P=\frac{1}{2}(B+B^T)\]?
