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 = `[(-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'
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
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.
For the matrix A = `[(1, 5),(6, 7)]`, verify that (A – A') is a skew symmetric matrix.
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Find the values of x, y, z if the matrix A = `[(0, 2y, z),(x, y, -z),(x, -y, z)]` satisfy the equation A'A = I.
Show that all the diagonal elements of a skew symmetric matrix are zero.
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.
If A is a square matrix, then AA is a
If A and B are symmetric matrices, then ABA 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
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
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)]`
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 and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
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 and B are matrices of same order, then (AB′ – BA′) is a ______.
Sum of two skew-symmetric matrices is always ______ matrix.
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 each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
If A, B are Symmetric matrices of same order, then AB – BA is a
Let A = `[(2, 3),(a, 0)]`, a ∈ R be written as P + Q where P is a symmetric matrix and Q is skew-symmetric matrix. If det(Q) = 9, then the modulus of the sum of all possible values of determinant of P is equal to ______.
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 ______.
For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?
If A and B are symmetric matrices of the same order, then AB – BA is ______.
