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' = `[(-2,3),(1,2)] and B = [(-1,0),(1,2)]` then find (A + 2B)'
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]`
Find `1/2` (A + A') and `1/2` (A -A') When `A = [(0, a, b),(-a,0,c),(-b,-c,0)]`
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 matrix A is both symmetric and skew-symmetric, then
If A is a square matrix, then AA is a
If A and B are symmetric matrices, then ABA is
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)]`
Show that A′A and AA′ are both symmetric matrices for any matrix A.
If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A–1 = A′, find value of α
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 ______.
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 and B are symmetric matrices, then AB – BA is a ______.
If A and B are symmetric matrices, then BA – 2AB 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 ______.
AA′ is always a symmetric matrix for any matrix A.
If A and B are symmetric matrices of the same order, then ____________.
If A and B are symmetric matrices of the same order, then ____________.
If A = `[(3, "x" - 1),(2"x" + 3, "x" + 2)]` is a symmetric matrix, then x = ____________.
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 ______.
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 ______.
