Show that the matrix B^{prime}AB is symmetric | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) We suppose that $A$ is a symmetric matrix,then $A^{\prime} = A$  ......... $(1)$Consider$(B^{\prime}AB)^{\prime} = \{B^{\prime}(AB)\}^{\prime}$$=

Vedclass · Accepted Answer

Vedclass · Answer

(A) We suppose that $A$ is a symmetric matrix,then $A^{\prime} = A$ ......... $(1)$
Consider
$(B^{\prime}AB)^{\prime} = \{B^{\prime}(AB)\}^{\prime}$
$= (AB)^{\prime}(B^{\prime})^{\prime}$ $[(AB)^{\prime} = B^{\prime}A^{\prime}]$
$= B^{\prime}A^{\prime}(B)$ $[(B^{\prime})^{\prime} = B]$
$= B^{\prime}(A^{\prime}B)$
$= B^{\prime}(AB)$ $[$Using $(1)]$
$\therefore (B^{\prime}AB)^{\prime} = B^{\prime}AB$
Thus,if $A$ is a symmetric matrix,then $B^{\prime}AB$ is a symmetric matrix.
Now,we suppose that $A$ is a skew-symmetric matrix,then $A^{\prime} = -A$ ......... $(2)$
Consider
$(B^{\prime}AB)^{\prime} = \{B^{\prime}(AB)\}^{\prime} = (AB)^{\prime}(B^{\prime})^{\prime}$
$= (B^{\prime}A^{\prime})B$
$= B^{\prime}(-A)B$ $[$Using $(2)]$
$= -B^{\prime}AB$
$\therefore (B^{\prime}AB)^{\prime} = -B^{\prime}AB$
Thus,if $A$ is a skew-symmetric matrix,then $B^{\prime}AB$ is a skew-symmetric matrix.
Hence,the matrix $B^{\prime}AB$ is symmetric or skew-symmetric according as $A$ is symmetric or skew-symmetric.

Show that the matrix $B^{\prime}AB$ is symmetric or skew-symmetric according as $A$ is symmetric or skew-symmetric.

Similar Questions

If $P$ is a $3 \times 3$ matrix such that $P^{\top}=2 P+I$,where $P^{\top}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix,then there exists a column matrix $X=\left[\begin{array}{l}x \\ y \\ z\end{array}\right] \neq\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ such that

If a square matrix $A$ is such that $\left(A^T-\frac{1}{2} I\right)\left(A-\frac{1}{2} I\right) = \left(A^T+\frac{1}{2} I\right)\left(A+\frac{1}{2} I\right) = I$,where $I$ is a unit matrix,then $A$ is

If $A = \begin{bmatrix} 3 & x-1 \\ 2x+3 & x+2 \end{bmatrix}$ is a symmetric matrix, then the value of $x$ is

If $A$ and $B$ are square matrices of the same order,then which of the following properties holds true for the transpose of their product?

If $A^{\prime}=\begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & 1 \end{bmatrix}$ and $B=\begin{bmatrix} -1 & 2 & 1 \\ 1 & 2 & 3 \end{bmatrix}$,then verify that $(A-B)^{\prime}=A^{\prime}-B^{\prime}$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module