Advertisements
Advertisements
प्रश्न
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1, 5),(-1, 2)]`
Advertisements
उत्तर
Let, A = `[(1, 5),(-1, 2)]`
⇒ `A' = [(1, -1),(5, 2)]`
`A + A' = [(1, 5),(-1, 2)] + [(1, -1),(5, 2)]`
= `[(1 + 1, 5 - 1),(-1 + 5, 2 + 2)]`
= `[(2, 4),(4, 4)]`
∴ `1/2 (A + A') = 1/2 [(2, 4),(4, 4)]`
= `[(1, 2),(2, 2)]`
And A – A' = `[(1, 5),(-1, 2)] - [(1, -1),(5, 2)]`
= `[(1 - 1, 5 + 1),(-1 - 5, 2 - 2)]`
= `[(0, 6),(-6, 0)]`
∴ `1/2 (A - A') = 1/2 [(0, 6),(-6, 0)] = [(0, 3),(-3, 0)]`
`A = 1/2 (A + A') + 1/2 (A - A')`
= `[(1, 2),(2, 2)] + [(0, 3),(-3, 0)]` = A
Symmetric matrices + Skew symmetric matrices
APPEARS IN
संबंधित प्रश्न
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 = `[(-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'
For the matrices A and B, verify that (AB)′ = B'A' where A = `[(0),(1),(2)]`, B = `[(1, 5, 7)]`
Find `1/2` (A + A') and `1/2` (A – A'), when A = `[(0, a, b),(-a, 0, c),(-b, -c, 0)]`
If the matrix A is both symmetric and skew symmetric, then ______.
If A and B are symmetric matrices of the same order, write whether AB − BA is symmetric or skew-symmetric or neither of the two.
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 matrix A is both symmetric and skew-symmetric, then
If A is a square matrix, then AA is a
If A and B are matrices of the same order, then ABT − BAT is a
The matrix \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a
Let A = `[(2, 3),(-1, 2)]`. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.
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 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.
The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` 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 is a symmetric matrix, then A3 is a ______ matrix.
If A is a skew-symmetric matrix, then A2 is a ______.
If A is symmetric matrix, then B′AB is ______.
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 = `[(3, "x" - 1),(2"x" + 3, "x" + 2)]` is a symmetric matrix, then x = ____________.
If A is any square matrix, then which of the following is skew-symmetric?
The diagonal elements of a skew symmetric matrix are ____________.
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 ______.
Which of the following is correct?
For a symmetric matrix \[A=[a_{ij}]_{n\times n}\], which entrywise relation is true for all \[i\] and \[j\]?
For a skew-symmetric matrix \[A=[a_{ij}]_{n\times n}\], which relation holds for all \[i\] and \[j\]?
If \[C=A-A^T\] for a square matrix \[A\], what is \[C^T\]?
