# Determine whether A is similar B .If A sim B, give an invertible matrix P such that P^{-1}AP = B A=begin{bmatrix}1 & 1&0 0&1&10&0&1 end{bmatrix} , B=begin{bmatrix}1&1&0 0&1&00&0&1 end{bmatrix}

Question
Matrices
Determine whether A is similar B .If $$A \sim B$$, give an invertible matrix P such that $$P^{-1}AP = B$$
$$A=\begin{bmatrix}1 & 1&0 \\0&1&1\\0&0&1 \end{bmatrix} , B=\begin{bmatrix}1&1&0 \\0&1&0\\0&0&1 \end{bmatrix}$$

2020-10-27
Step 1
The given matrices are $$A=\begin{bmatrix}1 & 1&0 \\0&1&1\\0&0&1 \end{bmatrix} \text{ and } B=\begin{bmatrix}1&1&0 \\0&1&0\\0&0&1 \end{bmatrix}$$
Here, it is observed that the matrices A and B are diagonal matrix.
Therefore, the eigenvalues are the entries on the main diagonal values.
Hence, the eigenvalues of both A, B are $$\lambda_1=1,\lambda_2=1 and \lambda_3=1$$
Step 2
Obtain the eigenvector corresponding to the eigenvalue $$\lambda=1$$ by solving the system (A-I)X=0. Let $$X=\begin{bmatrix}x_1 \\x_2 \\x_3 \end{bmatrix}$$
Then, $$\left(\begin{bmatrix}1 & 1&0 \\0&1&1\\0&0&1 \end{bmatrix}-\begin{bmatrix}1&1&0 \\0&1&0\\0&0&1 \end{bmatrix}\right)\begin{bmatrix}x_1 \\x_2 \\x_3 \end{bmatrix}=0$$
$$\begin{bmatrix}0&1&0 \\0&0&1\\0&0&0 \end{bmatrix}\begin{bmatrix}x_1 \\x_2 \\x_3 \end{bmatrix}=0$$
The system of equations corresponding to the above system is $$x_1=x_1$$(free variable)
$$x_2=0$$
$$x_3=0$$
If $$x_1=1 \text{ then } v_1= \begin{bmatrix}1 \\0 \\0 \end{bmatrix}$$
Step 3
Obtain the eigenvector corresponding to the eigenvalue $$\lambda=1$$ by solving the system (B-I)X=0.
Then $$\left(\begin{bmatrix}1 & 1&0 \\0&1&0\\0&0&1 \end{bmatrix}-\begin{bmatrix}1&0&0 \\0&1&0\\0&0&1 \end{bmatrix}\right)\begin{bmatrix}x_1 \\x_2 \\x_3 \end{bmatrix}=0$$
$$\begin{bmatrix}0&1&0 \\0&0&0\\0&0&0 \end{bmatrix}\begin{bmatrix}x_1 \\x_2 \\x_3 \end{bmatrix}=0$$
The system of equations corresponding to the above system is $$x_1=x_1$$(free variable)
$$x_2=0$$
$$x_3=x_3$$(free variable)
If $$x_1=1 , x_3=0 \text{ then } v_1= \begin{bmatrix}1 \\0 \\0 \end{bmatrix}$$
Choose the arbitrary value $$x_1=0 , x_3=1$$
Then the eigenvector is $$v_2= \begin{bmatrix}0 \\0 \\1 \end{bmatrix}$$
It is observed that the eigenspace corresponding to $$\lambda=1$$ of a matrix B has dimension 2 and the eigenspace corresponding to $$\lambda=1$$ of a matrix A has dimension 1.
Hence, the matrices are not similar.

### Relevant Questions

Use the matrix P to determine if the matrices A and A' are similar.
$$P=\begin{bmatrix}-1 & -1 \\1& 2 \end{bmatrix}, A=\begin{bmatrix}14 & 9 \\-20 & -13 \end{bmatrix} \text{ and } A'=\begin{bmatrix}3 & -2 \\2 & -2 \end{bmatrix}$$
$$P^{-1}=?$$
$$P^{-1}AP=?$$
Are they similar?
"Yes, they are similar" or "No, they are not similar"
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.
$$\begin{bmatrix}2 & 5&5 \\5 & 2&5\\5&5&2 \end{bmatrix}\lambda=-3.12$$
Find P and D
The row echelon form of a system of linear equations is given.
(a) Write the system of equations corresponding to the given matrix.
Use x, y, or x, y, z, or $$x_1,x_2,x_3, x_4$$
(b) Determine whether the system is consistent. If it is consistent, give the solution.
$$\begin{matrix}1 & 0 & 3 & 0 &1 \\ 0 & 1 & 4 & 3&2\\0&0&1&2&3\\0&0&0&0&0 \end{matrix}$$
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.
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) $$XF=F^{-1}-D^T$$
For each of the pairs of matrices that follow, determine whether it is possible to multiply the first matrix times the second. If it is possible, perform the multiplication.
$$\begin{bmatrix}1 & 4&3 \\0 & 1&4\\0&0&2 \end{bmatrix}\begin{bmatrix}3 & 2 \\1 & 1\\4&5 \end{bmatrix}$$
Find a family of the matrices that is similar to the matrix $$Q=\begin{bmatrix}p & -2 \\4 & -3 \end{bmatrix}$$
Use the given inverse of the coefficient matrix to solve the following system
$$5x_1+3x_2=6$$
$$-6x_1-3x_2=-2$$
$$A^{-1}=\begin{bmatrix}-1 & -1 \\2 & \frac{5}{3} \end{bmatrix}$$
Using givens rotation during QU factorization of the matrix A below, Make element (3,1) in A zero.
$$[A]=\begin{bmatrix}3 & 4 & 5 \\1 & 7 & 8 \\ 2 & 6 & 9\end{bmatrix}$$
$$A=\begin{bmatrix}2 & 1 & -1 \\-2 & 0 & 3 & \\ 2 & 1 & -4\\4 & 1 & -4 \\ 6 & 5 & -2\end{bmatrix}$$