Advertisements
Advertisements
प्रश्न
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)]`
Advertisements
उत्तर
We have A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`
Then A' = `[(2, 7, 1),(4, 3, -2),(-6, 5, 4)]`
Hence `("A" + "A'")/2 = 1/2 [(4, 11, -5),(11, 6, 3),(-5, 3, 8)]`
= `[(2, 11/2, (-5)/2),(11/2, 3, 3/2),((-5)/2, 3/2, 4)]`
and `("A" - "A'")/2 = 1/2 [(0, -3, -7),(3, 0, 7/2),(7, -7, 0)]`
= `[(0, (-3)/2, (-7)/2),(3/2, 0, 7/2),(7/2, (-7)/2, 0)]`
Therefore,
`("A" + "A'")/2 + ("A" - "A'")/2 = [(2, 11/2, (-5)/2),(11/2, 3, 3/2),((-5)/2, 3/2, 4)] + [(0, (-3)/2, (-7)/2),(3/2, 0, 7/2),(7/2, (-7)/2, 0)]`
= `[(2, 4, -6),(7, 3,5),(1,-2, 4)]`
= A
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 = `[(-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)]`
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 symmetric matrix.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 5),(1, -1)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 3, -1),(-2, -2, 1),(-4, -5, 2)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1, 5),(-1, 2)]`
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 ______.
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.
If a matrix A is both symmetric and skew-symmetric, then
If A and B are matrices of the same order, then ABT − BAT is a
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 ______.
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 `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
Sum of two skew-symmetric matrices is always ______ 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 A and B are any two matrices of the same order, then (AB)′ = A′B′.
If A is skew-symmetric matrix, then A2 is a symmetric matrix.
If A and B are symmetric matrices of the same order, then ____________.
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 = ____________.
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 ______.
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?
Which transpose property justifies the step \[(A+A^T)^T=A^T+(A^T)^T\]?
Which expression writes any square matrix \[A\] as the sum of a symmetric and a skew-symmetric matrix?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its symmetric part \[P=\frac{1}{2}(B+B^T)\]?
