# 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.

Let $A={I}_{2}$ and $B=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ . 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.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Alannej

Step 1
Given: $A={I}_{2}$ and $B=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$ Let $A=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
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 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.

Jeffrey Jordon