Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
\[Given: A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\]
\[ \Rightarrow A^T = \begin{bmatrix}1 & 0 \\ 2 & 3\end{bmatrix}\]
\[\text{Let B} = \frac{1}{2}\left( A + A^T \right) = \frac{1}{2}\left( \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 2 & 3\end{bmatrix} \right)\]
\[ = \frac{1}{2}\begin{bmatrix}1 + 1 & 2 + 0 \\ 0 + 2 & 3 + 3\end{bmatrix}\]
\[ = \frac{1}{2}\begin{bmatrix}2 & 2 \\ 2 & 6\end{bmatrix}\]
\[ = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix}\]
\[Now, \]
\[ B^T = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix} = B\]
\[ \text{Therefore, B is symmetric matrix }. \]
\[Let C = \frac{1}{2}\left( A - A^T \right) = \frac{1}{2}\left( \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix} - \begin{bmatrix}1 & 0 \\ 2 & 3\end{bmatrix} \right)\]
\[ = \frac{1}{2}\begin{bmatrix}1 - 1 & 2 - 0 \\ 0 - 2 & 3 - 3\end{bmatrix}\]
\[ = \frac{1}{2}\begin{bmatrix}0 & 2 \\ - 2 & 0\end{bmatrix}\]
\[ = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}\]
\[ \therefore C^T = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}^T = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix} = - \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} = C\]
\[So, \text{C is a skew - symmetric matrix }. \]
\[Now, \]
\[B + C = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix} + \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} = \begin{bmatrix}1 + 0 & 1 + 1 \\ 1 - 1 & 3 + 0\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix} = A\]
\[ \therefore B = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix}\]
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
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
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
Show that the matrix A = `[(0, 1, -1),(-1, 0, 1),(1, -1, 0)]` is a skew symmetric matrix.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(3, 5),(1, -1)]`
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.
For what value of x, is the matrix \[A = \begin{bmatrix}0 & 1 & - 2 \\ - 1 & 0 & 3 \\ x & - 3 & 0\end{bmatrix}\] a skew-symmetric matrix?
If a matrix A is both symmetric and skew-symmetric, then
If A and B are symmetric matrices, then ABA is
If A = [aij] is a square matrix of even order such that aij = i2 − j2, then
The matrix \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a
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)]`
Let A = `[(2, 3),(-1, 2)]`. Then show that A2 – 4A + 7I = O. Using this result calculate A5 also.
If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.
If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
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.
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric 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 any two matrices of the same order, then (AB)′ = A′B′.
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 and B are symmetric matrices of the same order, then ____________.
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 ______.
If `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q 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?
