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
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 = `[(-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 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, 5),(1, -1)]`
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1, 5),(-1, 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 ______.
Show that all the diagonal elements of a skew symmetric matrix are zero.
if A =`((5,a),(b,0))` is symmetric matrix show that a = b
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.
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
The matrix \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a
The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.
If A and B are matrices of same order, then (AB′ – BA′) is a ______.
If A is a symmetric matrix, then A3 is a ______ matrix.
If A is a skew-symmetric matrix, then A2 is a ______.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
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 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 ____________.
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 ______.
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 ______.
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?
For a symmetric matrix \[A=[a_{ij}]_{n\times n}\], which entrywise relation is true for all \[i\] and \[j\]?
For a skew-symmetric matrix \[A=[a_{ij}]_{n\times n}\], which relation holds for all \[i\] and \[j\]?
If \[C=A-A^T\] for a square matrix \[A\], what is \[C^T\]?
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 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?
