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.

Matrix transformations
asked 2020-12-25
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 \(AB=BA.\)

Answers (1)

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
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
\(A=\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix}\ and\ B=\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}\)
Then \(AB=\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix}\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}=\begin{bmatrix}0 & 1 \\0 & 0 \end{bmatrix}\)
and \(AB=\begin{bmatrix}0 & 0 \\0 & 1 \end{bmatrix}\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix}=\begin{bmatrix}0 & 0 \\1 & 0 \end{bmatrix}\)
so \(AB \neq BA\)

Relevant Questions

asked 2021-01-04
In the following question there are statements which are TRUE and statements which are FALSE.
Choose all the statements which are FALSE.
1. If the number of equations in a linear system exceeds the number of unknowns, then the system must be inconsistent - thus no solution.
2. If B has a column with zeros, then AB will also have a column with zeros, if this product is defined.
3. If AB + BA is defined, then A and B are square matrices of the same size/dimension/order.
4. Suppose A is an n x n matrix and assume A^2 = O, where O is the zero matrix. Then A = O.
5. If A and B are n x n matrices such that AB = I, then BA = I, where I is the identity matrix.
asked 2020-12-03
Show that if A and B are an n n matrices such that AB =O (where O is the zero matrix) then A is not invertible or B is not invertible.
asked 2020-11-24
Let n be a fixed positive integer greater thatn 1 and let a and b be positive integers. Prove that a mod n = b mon n if and only if a = b mod.
asked 2020-11-08
Prove that: If A or B is nonsingular, then AB is similar to BA
asked 2021-02-21
(7) If A and B are a square matrix of the same order. Prove that \(\displaystyle{\left({A}{B}{A}^{
asked 2021-03-02
Let T be the linear transformation from R2 to R2 consisting of reflection in the y-axis. Let S be the linear transformation from R2 to R2 consisting of clockwise rotation of 30◦. (b) Find the standard matrix of T , [T ]. If you are not sure what this is, see p. 216 and more generally section 3.6 of your text. Do that before you go looking for help!
asked 2021-01-24
It can be shown that the algebraic multiplicity of an eigenvalue lambda is always greater than or equal to the dimension of the eigenspace corresponding to lambda. Find h in the matrix A below such that the eigenspace for lambda = 5 is two-dimensional: \(\displaystyle{A}={\left[\begin{array}{cccc} {5}&-{2}&{6}&-{1}\\{0}&{3}&{h}&{0}\\{0}&{0}&{5}&{4}\\{0}&{0}&{0}&{1}\end{array}\right]}\)
asked 2021-01-25
Let D be the diagonal subset \(\displaystyle{D}={\left\lbrace{\left({x},{x}\right)}{\mid}{x}∈{S}_{{3}}\right\rbrace}\) of the direct product S_3 × S_3. Prove that D is a subgroup of S_3 × S_3 but not a normal subgroup.
asked 2020-12-14
Let G be a group. Let a,b,c denote elements of G, and let e be the nov element of G.
1. Prove that if ab=e, then ba=c (Hint: See theorem 2.
2. Prove that if abc=e, then cab=e and bca = e.
3. State a generalization of pants 1 and 2.
asked 2021-02-14
If A =[1,2,4,3], find B such that A+B=0