Advertisements
Advertisements
प्रश्न
If A = `[(sin alpha, cos alpha), (-cos alpha, sin alpha)]` then verify that A'A = I
उत्तर
Given, A = `[(sin alpha, cos alpha),(-cos alpha, sin alpha)]`
So, A' = `[(sin alpha, -cos alpha),(cos alpha, sin alpha)]`
Now, A' A = `[(sin alpha, -cos alpha),(cos alpha, sin alpha)] xx [(sin alpha, cos alpha),(-cos alpha, sin alpha)]`
`= [(sin^2 alpha+ cos^2 alpha, sin alpha cos alpha - cos alpha sin alpha),(cos alpha sin alpha - sin alpha cos alpha, cos^2 alpha + sin^2 alpha)]`
`= [(1, 0),(0,1)] = I` ... [Because `sin^2 alpha + cos^2 alpha = 1`]
Hence, it is proved that, A'A = I
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 =[(1),(-4), (3)], B = [-1, 2 1]`
Show that the matrix A = `[(1,-1,5),(-1,2,1),(5,1,3)]` is a symmetric matrix.
Show that the matrix A = `[(0,1,-1),(-1,0,1),(1,-1,0)]` is a skew symmetric matrix.
Find `1/2` (A + A') and `1/2` (A -A') When `A = [(0, a, b),(-a,0,c),(-b,-c,0)]`
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)]`
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.
Write a square matrix which is both symmetric as well as skew-symmetric.
If a matrix A is both symmetric and skew-symmetric, then
The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] is
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
If A and B are matrices of the same order, then ABT − BAT is a
If the matrix `((6,-"x"^2),(2"x"-15 , 10))` is symmetric, find the value of x.
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)]`
If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.
If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
______ matrix is both symmetric and skew-symmetric matrix.
Sum of two skew-symmetric matrices is always ______ 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 and B are symmetric matrices, then AB – BA is a ______.
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A is symmetric matrix, then B′AB is ______.
AA′ is always a symmetric matrix for any matrix A.
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 = `[(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?
If A = [aij] is a skew-symmetric matrix of order n, then ______.
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?
