Advertisements
Advertisements
प्रश्न
If A = `[(sin α, cos α), (-cos α, sin α)]`, then verify that A'A = I
Advertisements
उत्तर
Given, A = `[(sin α, cos α),(-cos α, sin α)]`
So, A' = `[(sin α, -cos α),(cos α, sin α)]`
Now, A' A = `[(sin α, -cos α),(cos α, sin α)] xx [(sin α, cos α),(-cos α, sin α)]`
= `[(sin^2 α + cos^2 α, sin α cos α - cos α sin α),(cos α sin α - sin α cos α, cos^2 α + sin^2 α)]`
= `[(1, 0),(0, 1)]` = I ...[∵ sin2 α + cos2 α = 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' = `[(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 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)]`
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.
If the matrix A is both symmetric and skew symmetric, then ______.
if A =`((5,a),(b,0))` is symmetric matrix show that a = b
If A is a square matrix, then AA is a
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
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 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.
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 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, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.
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 ______.
______ matrix is both symmetric and skew-symmetric matrix.
If A is a symmetric matrix, then A3 is a ______ 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 A and B are symmetric matrices, then BA – 2AB is a ______.
If A is skew-symmetric matrix, then A2 is a symmetric matrix.
If A = `[(3, "x" - 1),(2"x" + 3, "x" + 2)]` is a symmetric matrix, then x = ____________.
If A `= [(6,8,5),(4,2,3),(9,7,1)]` is the sum of a symmetric matrix B and skew-symmetric matrix C, then B is ____________.
The value of |A|, if A = `[(0, 2x - 1, sqrt(x)),(1 - 2x, 0, 2sqrt(x)),(-sqrt(x), -2sqrt(x), 0)]`, where x ∈ R+, is ______.
If A and B are symmetric matrices of the same order, then AB – BA is ______.
For a skew-symmetric matrix \[A=[a_{ij}]_{n\times n}\], which relation holds for all \[i\] and \[j\]?
What must be true of every diagonal element of a skew-symmetric matrix?
Which transpose property justifies the step \[(A+A^T)^T=A^T+(A^T)^T\]?
Which identity is used to replace \[(A^T)^T\] by \[A\] when proving that \[A+A^T\] is symmetric?
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 \[P=\frac{1}{2}(B+B^T)\]?
How many decompositions of a square matrix into a symmetric part and a skew-symmetric part are possible?
