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.
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 = [(-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'
if A' = `[(-2,3),(1,2)] and B = [(-1,0),(1,2)]` then find (A + 2B)'
If A = `[(sin alpha, cos alpha), (-cos alpha, sin alpha)]` then verify that A'A = I
For the matrix A = `[(1,5),(6,7)]` verify that (A + A') is a symmetric matrix.
Find `1/2` (A + A') and `1/2` (A -A') When `A = [(0, a, b),(-a,0,c),(-b,-c,0)]`
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)]`
If the matrix A is both symmetric and skew symmetric, then ______.
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 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
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.
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 and B are matrices of same order, then (AB′ – BA′) is a ______.
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 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 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 A = [aij] is a skew-symmetric matrix of order n, then ______.
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 ______.
The value of |A|, if A = `[(0, 2x - 1, sqrt(x)),(1 - 2x, 0, 2sqrt(x)),(-sqrt(x), -2sqrt(x), 0)]`, where x ∈ R+, is ______.
