# Let A=I_2 and B=begin{bmatrix}1 & 1 0 & 1 end{bmatrix} . Discuss the validity of the following statement and cite reasons for your conclusion: If A and B have the same trace , determinant , rank , and eigenvalues , then the matrices are similar.

Question
Matrices
Let $$A=I_2$$ and $$B=\begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}$$ . Discuss the validity of the following statement and cite reasons for your conclusion:
If A and B have the same trace , determinant , rank , and eigenvalues , then the matrices are similar.

2020-11-25
Step 1
Given: $$A=I_2$$ and $$B=\begin{bmatrix}1 & 1 \\0 & 1 \end{bmatrix}$$ Let $$A=\begin{bmatrix}1 & 0 \\0 & 1 \end{bmatrix}$$
Since the diagonal entries of matrix A are 1 therefore, the trace of the matrix A is 2. Also, the diagonal entries of matrix B are 1 therefore, the trace of the matrix B is also 2.
Step 2
Determinant of the matrices will be the product of diagonal entries. Therefore, the determinant of both the matrices is 1.
Rank of a matrix is the number of non zero rows. Therefore, the rank of matrix A and matrix B is 2.
The eigen values of a diagonal matrix is the elements in its main diagonal. Therefore, eigen values of matrix $$A is \lambda_1=1$$ and $$\lambda_2=1$$ Since, matrix B is an upper triangular matrix. Therefore, eigen values of matrix A is $$\lambda_1=1$$ and $$\lambda_2=1$$
Since, matrices A and B have the same trace, rank, determinant and eigen values therefore the matrices are similar.

### Relevant Questions

If $$A=\begin{bmatrix}-2 & 1&-4 \\-2 & 4&-1 \\ 1 &-1 &-4 \end{bmatrix} \text{ and } B=\begin{bmatrix}-2 & 4&2 \\-4 & -1&1 \\ 4 &1 &1 \end{bmatrix}$$
then AB=?
BA=?
True or false : AB=BA for any two square matrices A and B of the same size.
Given the matrix
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}$$
and suppose that we have the following row reduction to its PREF B
$$A=\begin{bmatrix}0 & 0&1 \\ 0 & 3&0 \\ 1&0 & -10 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 3&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&-10 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}\Rightarrow\begin{bmatrix}1 & 0&0 \\ 0 & 1&0 \\ 0&0 & 1 \end{bmatrix}$$
Write $$A \text{ and } A^{-1}$$ as product of elementary matrices.
Show that A and B are not similar matrices
$$A=\begin{bmatrix}1 & 0 &1 \\ 2 & 0 &2 \\3&0&3 \end{bmatrix} , B=\begin{bmatrix}1 & 1 &0 \\ 2 & 2 &0 \\0&1&1 \end{bmatrix}$$
Find the product of the following matrices , if possible
$$\begin{bmatrix}6 & -9 \\-5 & 5 \end{bmatrix}\begin{bmatrix}-1 \\6 \end{bmatrix}$$
Select the correct choise below and , if necessary , fill in the answer box to complete your choise.
A)$$\begin{bmatrix}6 & -9 \\-5 & 5 \end{bmatrix}\begin{bmatrix}-1 \\6 \end{bmatrix}$$
B)The product is not defined
Let $$A=\begin{bmatrix}1 & 0 \\ 0 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix}1 & 2 \\ 0 & 1 \end{bmatrix}$$ , show that A and B are not similar.
Consider the matrices
$$A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix},C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix},D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix} \text{ and } F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}$$
a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) $$AX^T=BC^3$$
(ii) $$A^{-1}(X-T)^T=(B^{-1})^T$$
(iii) $$XF=F^{-1}-D^T$$
Let $$A=\begin{bmatrix}1 & 2 \\4 & -1 \end{bmatrix} \text{ and } B=\begin{bmatrix}a & -3 \\-6& 1 \end{bmatrix}$$ . For what values of a (if any) do matrices A and B commute?
None
-2
-6
-1
-4
Compute the indicated matrices, if possible .
A^2B
let $$A=\begin{bmatrix}1 & 2 \\3 & 5 \end{bmatrix} \text{ and } B=\begin{bmatrix}2 & 0 & -1 \\3 & -3 & 4 \end{bmatrix}$$
$$P=\begin{bmatrix}-1 & -1 \\1& 2 \end{bmatrix}, A=\begin{bmatrix}14 & 9 \\-20 & -13 \end{bmatrix} \text{ and } A'=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}$$
$$P^{-1}=?$$
$$P^{-1}AP=?$$