Advertisements
Advertisements
Question
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3,3,-1),(-2,-2,1),(-4,-5,2)]`
Advertisements
Solution
A = `[(3,3,-1),(-2,-2,1),(-4,-5,2)]`
`=> A' = [(3,-2,-4),(3,-2,-5),(-1,1,2)]`
`therefore A + A' = [(3,3,-1),(-2,-2,1),(-4,-5,2)] + [(3,-2,-4),(3,-2,-5),(-1,-1,2)]`
`[(3 + 3, 3 - 2, -1 - 4),(-2 + 3, -2 -2, 1 -5),(-4 -1, -5 + 1, 2 + 2)]`
`= [(6,1,-5),(1,-4,-4),(-5,-4,4)]`
`therefore 1/2 (A + A')`
`= 1/2 [(6,1,-5),(1,-4,-4),(-5,-4,4)]`
`= [(3,1/2,-5/2),(1/2,-2,-2),(-5/2,-2,2)]`
and A - A' `= [(3,3,-1),(-2,-2,1),(-4,-5,2)] - [(3,-2,-4),(3,-2,-5),(-1,-1,2)]`
`= [(3 - 3, 3 + 2, -1 + 4),(-2 - 3, -2 + 2, 1 + 5),(-4 +1, -5 - 1, 2 - 2)]`
`= [(0,5,3),(-5,0,6),(-3,-6,0)]`
`1/2 (A - A') = 1/2 [(0,5,3),(-5,0,6),(-3,-6,0)]`
`= [(0,5/2,3/2),(-5/2,0,3),(-3/2,-3,0)]`
`A = 1/2 (A + A') + 1/2 (A - A')`
`= [(3,1/2,-5/2),(1/2,-2,-2),(-5/2,-2,2)] + [(0,5/2,3/2),(-5/2,0,3),(-3/2,-3,0)]`
`= [(3 + 0, 1/2 + 5/2, -5/2 + 3/2),(1/2 - 5/2, -2 + 0, -2 + 3),(-5/2 - 3/2, -2 -3, 2 + 0)]`
`= [(3,3,-1),(-2,-2,1),(-4,-5,2)] = A`
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 is a skew symmetric matric of order 3, then prove that det A = 0
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]`
If A = `[(cos alpha, sin alpha), (-sin alpha, cos alpha)]` then verify that A' A = I
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.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3,5),(1,-1)]`
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
If the matrix A is both symmetric and skew symmetric, then ______.
if A =`((5,a),(b,0))` is symmetric matrix show that a = b
Write a square matrix which is both symmetric as well as skew-symmetric.
If \[A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\] is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.
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
The matrix \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is
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 = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
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.
If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.
If A and B are matrices of same order, then (AB′ – BA′) is a ______.
If A is skew-symmetric, then kA is a ______. (k is any scalar)
If A and B are symmetric matrices, then AB – BA is a ______.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
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 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 ____________.
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 ______.
Let A and B be and two 3 × 3 matrices. If A is symmetric and B is skewsymmetric, then the matrix AB – BA is ______.
Number of symmetric matrices of order 3 × 3 with each entry 1 or – 1 is ______.
