Advertisements
Advertisements
प्रश्न
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
विकल्प
\[\begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}\]
\[\begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[\begin{bmatrix}4 & 4 & - 8 \\ 4 & 6 & 8 \\ - 8 & 8 & 4\end{bmatrix}\]
\[\begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}\]
उत्तर
\[\begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}\]
\[Here, \]
\[ A = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix}\]
\[ \Rightarrow A^T = \begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[Now, \]
\[A + A^T = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix} + \begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[ \Rightarrow A + A^T = \begin{bmatrix}2 + 2 & 0 + 4 & - 3 - 5 \\ 4 + 0 & 3 + 3 & 1 + 7 \\ - 5 - 3 & 7 + 1 & 2 + 2\end{bmatrix}\]
\[ \Rightarrow A + A^T = \begin{bmatrix}4 & 4 & - 8 \\ 4 & 6 & 8 \\ - 8 & 8 & 4\end{bmatrix}\]
\[A - A^T = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix} - \begin{bmatrix}2 & 4 & - 5 \\ 0 & 3 & 7 \\ - 3 & 1 & 2\end{bmatrix}\]
\[ \Rightarrow A - A^T = \begin{bmatrix}2 - 2 & 0 - 4 & - 3 + 5 \\ 4 - 0 & 3 - 3 & 1 - 7 \\ - 5 + 3 & 7 - 1 & 2 - 2\end{bmatrix}\]
\[ \Rightarrow A - A^T = \begin{bmatrix}0 & - 4 & 2 \\ 4 & 0 & - 6 \\ - 2 & 6 & 0\end{bmatrix}\]
\[\text{Let P }= \frac{1}{2}\left( A + A^T \right) = \frac{1}{2}\begin{bmatrix}4 & 4 & - 8 \\ 4 & 6 & 8 \\ - 8 & 8 & 4\end{bmatrix} = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}\]
\[Q = \frac{1}{2}\left( A - A^T \right) = \frac{1}{2}\begin{bmatrix}0 & - 4 & 2 \\ 4 & 0 & - 6 \\ - 2 & 6 & 0\end{bmatrix} = \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix}\]
\[Now, \]
\[ P^T = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix}^T = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix} = P\]
\[ Q^T = \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix}^T = \begin{bmatrix}0 & 2 & - 1 \\ - 2 & 0 & 3 \\ 1 & - 3 & 0\end{bmatrix} = - \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix} = - Q\]
Thus, P is symmetric and Q is skew - symmetric .
\[ P + Q = \begin{bmatrix}2 & 2 & - 4 \\ 2 & 3 & 4 \\ - 4 & 4 & 2\end{bmatrix} + \begin{bmatrix}0 & - 2 & 1 \\ 2 & 0 & - 3 \\ - 1 & 3 & 0\end{bmatrix}\]
\[ = \begin{bmatrix}2 + 0 & 2 - 2 & - 4 + 1 \\ 2 + 2 & 3 + 0 & 4 - 3 \\ - 4 - 1 & 4 + 3 & 2 + 0\end{bmatrix}\]
\[ = \begin{bmatrix}2 & 0 & - 3 \\ 4 & 3 & 1 \\ - 5 & 7 & 2\end{bmatrix} = A\]
Thus, we have expressed A is the sum of a symmetric and a skew - symmetric matrix .
Hence, the symmetric matrix is`[[ 2 2 - 4 ],[ 2 3 4],[ - 4 4 2]]`
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' [(3,4),(-1, 2),(0,1)] and B = [((-1,2,1),(1,2,3))]` 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 matrix A = `[(1,5),(6,7)]` verify that (A + A') is a symmetric matrix.
Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
`[(1,5),(-1,2)]`
If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.
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 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.
The matrix \[\begin{bmatrix}0 & 5 & - 7 \\ - 5 & 0 & 11 \\ 7 & - 11 & 0\end{bmatrix}\] 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
If A and B are matrices of the same order, then ABT − BAT is a
The matrix \[A = \begin{bmatrix}0 & - 5 & 8 \\ 5 & 0 & 12 \\ - 8 & - 12 & 0\end{bmatrix}\] is a
The matrix \[A = \begin{bmatrix}1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4\end{bmatrix}\] is
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 = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A2 – B2
Show that A′A and AA′ are both symmetric matrices for any matrix A.
Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.
The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.
The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.
If A and B are symmetric matrices, then AB – BA is a ______.
If A is symmetric matrix, then B′AB is ______.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if ______.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
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 A is any square matrix, then which of the following is skew-symmetric?
If A, B are Symmetric matrices of same order, then AB – BA is a
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 `[(2, 0),(5, 4)]` = P + Q, where P is symmetric, and Q is a skew-symmetric matrix, then Q is equal to ______.
