Advertisements
Advertisements
प्रश्न
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3,3,-1),(-2,-2,1),(-4,-5,2)]`
उत्तर
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
संबंधित प्रश्न
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]`
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.
For the matrix A = `[(1,5),(6,7)]` verify that (A + A') is a symmetric matrix.
For the matrix A = `[(1,5),(6,7)]` verify that (A - A') is a skew symmetric matrix.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1,5),(-1,2)]`
Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.
Show that all the diagonal elements of a skew symmetric matrix are zero.
if A =`((5,a),(b,0))` is symmetric matrix show that a = b
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.
The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] is
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
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
The matrix \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is
If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.
Let A = `[(2, 3),(-1, 2)]`. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.
If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.
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 skew-symmetric matrix, then A2 is a ______.
If A is skew-symmetric, then kA is a ______. (k is any scalar)
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
AA′ is always a symmetric matrix for any matrix A.
If A and B are symmetric matrices of the same order, then ____________.
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 ______.
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?
