Advertisements
Advertisements
Question
Write a square matrix which is both symmetric as well as skew-symmetric.
Advertisements
Solution
\[Let A = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} \]
\[ A^T = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
`"Since" A^T = A, A is a symmmetric matrix `
\[Now, \]
\[ - A = - \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix} \]
\[ \Rightarrow - A = \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}\]
`"Since" A^T = - A, A is a skew - symmetric matrix . `
Thus,` A= [[0 0 ],[0 0]] `is an example of a matrix that is both symmetric and skew - symmetric.
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' = `[(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 = `[(0),(1),(2)]`, B = `[(1, 5, 7)]`
Show that the matrix A = `[(0, 1, -1),(-1, 0, 1),(1, -1, 0)]` is a skew symmetric matrix.
For the matrix A = `[(1, 5),(6, 7)]`, verify that (A – A') is a skew symmetric matrix.
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
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.
Show that all the diagonal elements of a skew symmetric matrix are zero.
If A and B are symmetric matrices of the same order, write whether AB − BA is symmetric or skew-symmetric or neither of the two.
For what value of x, is the matrix \[A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\] a skew-symmetric matrix?
If \[A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\] is expressed as the sum of a symmetric and skew-symmetric matrix, then the symmetric matrix is
If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.
Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`
If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
Show that A′A and AA′ are both symmetric matrices for any matrix A.
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 ______.
Sum of two skew-symmetric matrices is always ______ matrix.
If A is skew-symmetric, then kA is a ______. (k is any scalar)
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A and B are any two matrices of the same order, then (AB)′ = A′B′.
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?
If A = [aij] is a skew-symmetric matrix of order n, then ______.
Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is ______.
If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to ______.
Which of the following is correct?
A square matrix \[A=[a_{ij}]_{n\times n}\] is symmetric when which condition holds?
For a symmetric matrix \[A=[a_{ij}]_{n\times n}\], which entrywise relation is true for all \[i\] and \[j\]?
For a skew-symmetric matrix \[A=[a_{ij}]_{n\times n}\], which relation holds for all \[i\] and \[j\]?
What must be true of every diagonal element of a skew-symmetric matrix?
Which transpose property justifies the step \[(A+A^T)^T=A^T+(A^T)^T\]?
How many decompositions of a square matrix into a symmetric part and a skew-symmetric part are possible?
