Advertisements
Advertisements
प्रश्न
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
Advertisements
उत्तर
Here, A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`
Given that: A–1 = A′
Pre-multiplying both sides by A
AA–1 = AA′
⇒ I = AA′ ......[∵ AA–1 = I]
⇒ `[(1, 0),(0, 1)] = [(cosalpha, sinalpha),(-sinalpha, cosalpha)] [(cosalpha, - sinalpha),(sinalpha, cosalpha)]`
⇒ `[(1, 0),(0, 1)] = [(cos^2alpha + sin^2alpha, -sinalpha cosalpha + sinalpha cosalpha),(-sinalpha cosalpha + cosalpha sinalpha, sin^2alpha + cos^2alpha)]`
⇒ `[(1, 0),(0, 1)] = [(1, 0),(0, 1)]`
Hence, it is true for all values of 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' = `[(-2, 3),(1, 2)]` and B = `[(-1, 0),(1, 2)]`, then find (A + 2B)'
For the matrices A and B, verify that (AB)′ = B'A' where A = `[(0),(1),(2)]`, B = `[(1, 5, 7)]`
If A = `[(sin α, cos α), (-cos α, sin α)]`, then verify that A'A = I
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.
For the matrix A = `[(1, 5),(6, 7)]`, verify that (A – A') 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)]`
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.
The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] is
If A is a square matrix, then AA is a
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 and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
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 ______.
Sum of two skew-symmetric matrices is always ______ matrix.
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 P is of order 2 x 3 and Q is of order 3 x 2, then PQ is of order ____________.
If A is any square matrix, then which of the following is skew-symmetric?
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 ____________.
If A, B are Symmetric matrices of same order, then AB – BA is a
If A = [aij] is a skew-symmetric matrix of order n, then ______.
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 ______.
For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?
Which of the following is correct?
If \[C=A-A^T\] for a square matrix \[A\], what is \[C^T\]?
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?
Why does multiplying \[A+A^T\] by \[\frac{1}{2}\] not change its symmetric property?
For \[B=\begin{bmatrix}2&-2&-4\\-1&3&4\\1&-2&-3\end{bmatrix}\], what is its skew-symmetric part \[Q=\frac{1}{2}(B-B^T)\]?
For the matrices \[P=\frac{1}{2}(B+B^T)\] and \[Q=\frac{1}{2}(B-B^T)\], what is \[P+Q\]?
