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
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 = `[(1, -1, 5),(-1, 2, 1),(5, 1, 3)]` is a symmetric matrix.
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 symmetric matrix.
Find `1/2` (A + A') and `1/2` (A – A'), when A = `[(0, a, b),(-a, 0, c),(-b, -c, 0)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
If the matrix A is both symmetric and skew symmetric, then ______.
Show that all the diagonal elements of a skew symmetric matrix are zero.
if A =`((5,a),(b,0))` is symmetric matrix show that a = b
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
The matrix \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a
Show that a matrix which is both symmetric and skew symmetric is a zero matrix.
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 symmetric matrices of the same order, then (AB′ –BA′) is a ______.
If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.
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 ______.
______ matrix is both symmetric and skew-symmetric matrix.
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A is symmetric matrix, then B′AB is ______.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
If A and B are any two matrices of the same order, then (AB)′ = A′B′.
If A and B are symmetric matrices of the same order, then ____________.
If A `= [(6,8,5),(4,2,3),(9,7,1)]` is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is ____________.
Let A = `[(2, 3),(a, 0)]`, a ∈ R be written as P + Q where P is a symmetric matrix and Q is skew-symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to ______.
If ax4 + bx3 + cx2 + dx + e = `|(2x, x - 1, x + 1),(x + 1, x^2 - x, x - 1),(x - 1, x + 1, 3x)|`, then the value of e is ______.
Number of symmetric matrices of order 3 × 3 with each entry 1 or – 1 is ______.
For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?
