Advertisements
Advertisements
प्रश्न
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3,5),(1,-1)]`
Advertisements
उत्तर
to suppose, A = `[(3,5),(1, -1)],` A' =`[(3,1),(5, -1)]`
So, A `1/2` (A + A') + `1/2` (A - A')
Let, P = `1/2` (A + A') = `1/2 ([(3, 5),(1, -1)]) + ([(3, 1),(5, -1)])`
`= 1/2 [(3 + 3, 5 + 1), (1 + 5, -1 -1)]`
`= 1/2 [(6, 6), (6, -2)]`
`= [(3, 3), (3, -1)]`
and, `"P'" = [(3, 3), (3, -1)] = "P",`
Therefore, the matrix P is a symmetric matrix.
Then, Q = `1/2` (A - A') = `1/2 ([(3, 5),(1, -1)]) - ([(3, 1),(5, -1)])`
`= 1/2 [(3 - 3, 5 -1), (1 - 5, -1 + 1)]`
`= 1/2 [(0,4), (-4, 0)]`
`= [(0,2), (-2, 0)]`
and, Q' = `[(0,2), (-2, 0)]` = - Q,
Hence the matrix Q is a skew symmetric matrix.
So, A = P + Q
= `[(3, 3), (3, -1)] + [(0, -2),(-2, 0)]`
APPEARS IN
संबंधित प्रश्न
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' [(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'
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.
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 ______.
if A =`((5,a),(b,0))` is symmetric matrix show that a = b
If a matrix A is both symmetric and skew-symmetric, then
The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] is
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 \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a
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.
Let A = `[(2, 3),(-1, 2)]`. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.
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
The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
______ matrix is both symmetric and skew-symmetric matrix.
If A is a symmetric matrix, then A3 is a ______ matrix.
If A is a skew-symmetric matrix, then A2 is a ______.
If A and B are any two matrices of the same order, then (AB)′ = A′B′.
AA′ is always a symmetric matrix for any matrix A.
If A = `[(3, "x" - 1),(2"x" + 3, "x" + 2)]` is a symmetric matrix, then x = ____________.
The diagonal elements of a skew symmetric matrix are ____________.
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 ______.
The value of |A|, if A = `[(0, 2x - 1, sqrt(x)),(1 - 2x, 0, 2sqrt(x)),(-sqrt(x), -2sqrt(x), 0)]`, where x ∈ R+, is ______.
If A and B are symmetric matrices of the same order, then AB – BA is ______.
