Advertisements
Advertisements
Question
If A = [aij] is a skew-symmetric matrix of order n, then ______.
Options
`a_(ij) = 1/(a_(ji)) ∀ i, j`
`a_(ij) ≠ 0 ∀ i, j`
`a_(ij) = 0, where i = j`
`a_(ij) ≠ 0 where i = j`
Advertisements
Solution
If A = [aij] is a skew-symmetric matrix of order n, then `underline(bb(a_(ij) = 0, where i = j))`.
Explanation:
In a skew-symmetric matrix, the (i, j)th element is negative of the (j, i)th element. Hence, the (i, i)th element = 0
APPEARS IN
RELATED QUESTIONS
Matrix A = `[(0,2b,-2),(3,1,3),(3a,3,-1)]`is given to be symmetric, find values of a and b
If A= `((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.
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' = `[(-2, 3),(1, 2)]` and B = `[(-1, 0),(1, 2)]`, then find (A + 2B)'
For the matrices A and B, verify that (AB)′ = B'A' where A = `[(0),(1),(2)]`, B = `[(1, 5, 7)]`
Show that the matrix A = `[(1, -1, 5),(-1, 2, 1),(5, 1, 3)]` is a symmetric matrix.
For the matrix A = `[(1, 5),(6, 7)]`, verify that (A – A') is a skew symmetric matrix.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
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 A = [aij] is a square matrix of even order such that aij = i2 − j2, then
If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.
Show that a matrix which is both symmetric and skew symmetric is a zero matrix.
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 the matrix `[(0, "a", 3),(2, "b", -1),("c", 1, 0)]`, is a skew symmetric matrix, find the values of a, b and c.
The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.
The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
If A and B are matrices of same order, then (AB′ – BA′) is a ______.
AA′ is always a symmetric matrix for any matrix A.
If A is skew-symmetric matrix, then A2 is a symmetric matrix.
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?
For a skew-symmetric matrix \[A=[a_{ij}]_{n\times n}\], which relation holds for all \[i\] and \[j\]?
Which identity is used to replace \[(A^T)^T\] by \[A\] when proving that \[A+A^T\] is symmetric?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its symmetric part \[P=\frac{1}{2}(B+B^T)\]?
How many decompositions of a square matrix into a symmetric part and a skew-symmetric part are possible?
