Advertisements
Advertisements
प्रश्न
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
Advertisements
उत्तर
A = `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
⇒ A = `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
∴ A + A' = `[(6,-2,2),(-2,3,-1),(2,-1,3)] + [(6,-2,2),(-2,3,-1),(2,-1,3)]`
= `[(6 + 6, -2 - 2, 2 + 2),(-1 - 1, 3 + 3, -1 - 1),(2 + 2, -1 - 1, 3 + 3)]`
= `[(12, -4, 4),(-4, 6, -2),(4, -2, 6)]`
∴ `1/2 (A + A') = 1/2 [(12, -4, 4),(-4, 6, -2),(4, -2, 6)]`
= `[(6, -2, 2),(-2, 3, -1),(4, -1, 3)]` is a symmetric matrix.
∴ (A – A) = `[(6, -2, 2),(-2, 3, -1),(4, -1, 3)] - [(6, -2, 2),(-2, 3, -1),(4, -1, 3)]`
= `[(0, 0, 0),(0, 0, 0),(0, 0, 0)]`
∴ `1/2 (A - A') + 1/2 [(0, 0, 0),(0, 0, 0),(0, 0, 0)] = 0`
Hence, `A = 1/2 (A + A') + 1/2 (A - A')`
= `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)] + [(0, 0, 0),(0, 0, 0),(0, 0, 0)]`
= `[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]` = A
APPEARS IN
संबंधित प्रश्न
If A is a skew symmetric matric of order 3, then prove that det A = 0
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'
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 = `[(sin α, cos α), (-cos α, sin α)]`, then verify that A'A = I
Show that the matrix A = `[(1, -1, 5),(-1, 2, 1),(5, 1, 3)]` is a symmetric matrix.
For the matrix A = `[(1, 5),(6, 7)]`, verify that (A – A') is a skew symmetric matrix.
If A and B are symmetric matrices, prove that AB – BA is a skew symmetric matrix.
If the matrix A is both symmetric and skew symmetric, then ______.
Write a square matrix which is both symmetric as well as skew-symmetric.
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?
The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] is
If A is a square matrix, then AA is a
If A and B are symmetric matrices, then ABA is
If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B 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.
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 = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
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.
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 is a skew-symmetric matrix, then A2 is a ______.
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A is symmetric matrix, then B′AB is ______.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
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 ______.
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 ______.
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 A and B are symmetric matrices of the same order, then AB – BA is ______.
