Advertisements
Advertisements
Question
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.
Solution
We have, A = `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]`
We know that A = `("A" + "A'")/2 + ("A" - "A'")/2`
Where `("A" + "A'")/2` is symmetric and `("A" - "A'")/2` is skew-symmetric
∴ A' = `[(2, 1, 4),(3, -1, 1),(1, 2, 2)]`
Now, `("A" + "A'")/2 = ([(2, 3, 1),(1, -1, 2),(4, 1, 2)] + [(2, 1, 4),(3, -1, 1),(1, 2, 2)])/2`
= `1/2 [(4, 4, 5),(4, -2, 3),(5, 3, 4)]`
= `[(2, 2, 5/2),(2, -1, 3/2),(5/2, 3/2, 2)]`
And `("A" - "A'")/2 = ([(2, 3, 1),(1, -1, 2),(4, 1, 2)] - [(2, 1, 4),(3, -1, 1),(1, 2, 2)])/2`
= `1/2 [(0, 2, -3),(-2, 0, 1),(3, -1, 0)]`
= `[(0,1, (-3)/2),(-1, 0, 1/2),(3/2, (-1)/2, 0)]`
∴ A = `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]`
= `[(2, 2, 5/2),(2, -1, 3/2),(5/2, 3/2, 2)] + [(0, 1, (-3)/2),(-1, 0, 1/2),(3/2, 1/2, 0)]`
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' [(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'
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 =[(1),(-4), (3)], B = [-1, 2 1]`
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.
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:
`[(3,5),(1,-1)]`
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
If A is a square matrix, then AA is a
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
If A and B are matrices of the same order, then ABT − BAT is a
The matrix \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is
If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.
If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
If A is skew-symmetric, then kA is a ______. (k is any scalar)
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.
AA′ is always a symmetric matrix for any matrix A.
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 ______.
For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?
