Prove that AB is symmetric if and only

b) Find symmetric

matrices A and B such that

Phoebe
2020-12-25
Answered

a) Let A and B be symmetric matrices of the same size.

Prove that AB is symmetric if and only$AB=BA.$

b) Find symmetric$2\cdot 2$

matrices A and B such that$AB=BA.$

Prove that AB is symmetric if and only

b) Find symmetric

matrices A and B such that

You can still ask an expert for help

Layton

Answered 2020-12-26
Author has **89** answers

a) $\Rightarrow $ Suppose that AB is symmetric.

This means that$(AB{)}^{T}=AB$

Since$(AB{)}^{T}={B}^{T}{A}^{T}=BA$

(because A and B are symmetric), we get$AB=BA$

as required

$\Leftarrow $

Suppose that$Ab=BA$

To prove that AB is symmetric, we will prove that$AB=(AB{)}^{T}.$

Since$(AB{)}^{T}={B}^{T}{A}^{T}=BA=AB$

where we used that A and B are symmetric in the second equality and the assumption$Ba=Ab$

in the last equation and the assumption$BA=BA$ in the last equality,

we get that$(AB)T=AB$

as required

b) Let

$A=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]\text{}and\text{}B=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$

Then$AB=\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$

and$AB=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]$

so$AB\ne BA$

This means that

Since

(because A and B are symmetric), we get

as required

Suppose that

To prove that AB is symmetric, we will prove that

Since

where we used that A and B are symmetric in the second equality and the assumption

in the last equation and the assumption

we get that

as required

b) Let

Then

and

so

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My attempt: So I'm guessing there exists a matrix (a transformation matrix) and it must be of order (2,3) for it to give (-2,2,-7)${}^{T}$ when multiplied by (1,3)${}^{T}$:

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