Advertisements
Advertisements
Question
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
Solution
\[Given: A = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix}\]
\[ \Rightarrow A^T = \begin{bmatrix}1 & 0 \\ 2 & 3\end{bmatrix}\]
\[\text{Let B} = \frac{1}{2}\left( A + A^T \right) = \frac{1}{2}\left( \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 2 & 3\end{bmatrix} \right)\]
\[ = \frac{1}{2}\begin{bmatrix}1 + 1 & 2 + 0 \\ 0 + 2 & 3 + 3\end{bmatrix}\]
\[ = \frac{1}{2}\begin{bmatrix}2 & 2 \\ 2 & 6\end{bmatrix}\]
\[ = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix}\]
\[Now, \]
\[ B^T = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix} = B\]
\[ \text{Therefore, B is symmetric matrix }. \]
\[Let C = \frac{1}{2}\left( A - A^T \right) = \frac{1}{2}\left( \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix} - \begin{bmatrix}1 & 0 \\ 2 & 3\end{bmatrix} \right)\]
\[ = \frac{1}{2}\begin{bmatrix}1 - 1 & 2 - 0 \\ 0 - 2 & 3 - 3\end{bmatrix}\]
\[ = \frac{1}{2}\begin{bmatrix}0 & 2 \\ - 2 & 0\end{bmatrix}\]
\[ = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}\]
\[ \therefore C^T = \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix}^T = \begin{bmatrix}0 & - 1 \\ 1 & 0\end{bmatrix} = - \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} = C\]
\[So, \text{C is a skew - symmetric matrix }. \]
\[Now, \]
\[B + C = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix} + \begin{bmatrix}0 & 1 \\ - 1 & 0\end{bmatrix} = \begin{bmatrix}1 + 0 & 1 + 1 \\ 1 - 1 & 3 + 0\end{bmatrix} = \begin{bmatrix}1 & 2 \\ 0 & 3\end{bmatrix} = A\]
\[ \therefore B = \begin{bmatrix}1 & 1 \\ 1 & 3\end{bmatrix}\]
APPEARS IN
RELATED QUESTIONS
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'
For the matrices A and B, verify that (AB)′ = B'A', where A = `[(1),(-4),(3)]`, B = `[(-1, 2, 1)]`
Show that the matrix A = `[(0, 1, -1),(-1, 0, 1),(1, -1, 0)]` is a skew 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:
`[(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 A =`((5,a),(b,0))` is symmetric matrix show that a = b
Write a square matrix which is both symmetric as well as skew-symmetric.
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
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
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)]`
If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.
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 a symmetric matrix, then A3 is a ______ matrix.
If A and B are symmetric matrices, then AB – BA is a ______.
If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
AA′ is always a symmetric matrix for any matrix A.
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 and B are symmetric matrices of the same order, then ____________.
The diagonal elements of a skew symmetric matrix are ____________.
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 ______.
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 ______.
For what value of k the matrix `[(0, k),(-6, 0)]` is a skew symmetric matrix?
If A and B are symmetric matrices of the same order, then AB – BA is ______.
