Advertisements
Advertisements
प्रश्न
Show that A′A and AA′ are both symmetric matrices for any matrix A.
Advertisements
उत्तर
Let P = A'A
⇒ P' = (A'A)'
⇒ P' = A'(A')' .....[(AB') = B'A']
⇒ P' = A'A ......[∵ (A')' = A]
⇒ P' = P
Hence, A'A is a symmetric matrix.
Now, Let Q = AA'
⇒ Q' = (AA')'
⇒ Q' = (A')A' .....[(AB)' = B'A']
⇒ Q' = AA' ......[∵ (A')' = A]
⇒ Q' = Q
Hence, AA' is also a symmetric matrix.
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 is a skew symmetric matric of order 3, then prove that det A = 0
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' = `[(-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 = `[(cos α, sin α), (-sin α, cos α)]`, 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:
`[(6, -2, 2),(-2, 3, -1),(2, -1, 3)]`
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 =`((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.
If a matrix A is both symmetric and skew-symmetric, then
If A and B are symmetric matrices, then ABA is
If A and B are two matrices of order 3 × m and 3 × n respectively and m = n, then the order of 5A − 2B is
The matrix \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] 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.
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
Sum of two skew-symmetric matrices is always ______ matrix.
If A is skew-symmetric, then kA is a ______. (k is any scalar)
If A and B are symmetric matrices, then BA – 2AB is a ______.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
AA′ is always a symmetric matrix for any matrix A.
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 diagonal elements of a skew symmetric matrix are ____________.
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 ______.
If A and B are symmetric matrices of the same order, then AB – BA is ______.
Which of the following is correct?
