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
asked 2020-11-24
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.

Answers (1)

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.
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