Advertisements
Advertisements
प्रश्न
if `A' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))]` then verify that (A + B)' = A' + B'
उत्तर
Given, A' = `[(3,4),(-1,2),(0,1)]` and B = `[(-1,2,1),(1,2,3)]`
Then, A = `[(3, -1, 0),(4,2,1)]` and B' = `[(-1,1),(2,2),(1,3)]` [Because(A)' = A]
Now, (A + B) = `[(3, -1, 0),(4,2,1)] + [(-1,2,1),(1,2,3)]`
`= [(3 - 1, -1 + 2, 0 + 1),(4 + 1, 2 + 2, 1 + 3)]`
`= [(2,1,1),(5,4,4)]`
Then, (A + B)' = `[(2,5),(1,4),(1,4)]` ....(i)
A' + B' = `[(3,4),(-1,2),(0,1)] + [(-1,1),(2,2),(1,3)]`
`= [(3 - 1, 4 + 1),(-1 + 2, 2 + 2), (0 + 1, 1 + 3)]`
`= [(2,5),(1,4),(1,4)]` ...(ii)
Equations (i) and (ii) prove that,
(A + B)' = A' + B'
APPEARS IN
संबंधित प्रश्न
If A= `((3,5),(7,9))`is written as A = P + Q, where P is a symmetric matrix and Q is skew symmetric matrix, then write the matrix P.
If A is a skew symmetric matric of order 3, then prove that det A = 0
If A = `[(cos alpha, sin alpha), (-sin alpha, cos alpha)]` then verify that A' A = I
If A = `[(sin alpha, cos alpha), (-cos alpha, sin alpha)]` then verify that A'A = I
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:
`[(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.
Show that all the diagonal elements of a skew symmetric matrix are zero.
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 \[A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\] is written as B + C, where B is a symmetric matrix and C is a skew-symmetric matrix, then B is equal to.
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 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
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 ______.
Show that A′A and AA′ are both symmetric matrices for any matrix 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.
The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.
The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
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 and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
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 ______.
If A and B are symmetric matrices of the same order, then AB – BA is ______.
