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.

tabita57i 2020-11-08 Answered
Let A=[1001] and B=[1201] , show that A and B are not similar.
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Expert Answer

gotovub
Answered 2020-11-09 Author has 98 answers
Step 1
To check the similarity of the matrices, calculate the determinant of the matrices A=[1001] and B=[1201]
|A|=|1001|
=10
=1
|B|=|1201|
=10
=1
Here, the determinants of the matrices A and B are equal.
Step 2
Now, find the eigen values of the matrices A and B.
|AλI|=0
|[1001]λ[1001]|=0
|1λ001λ|=0
(1λ)2=0
1λ=0
λ1=1
λ2=1
|BλI|=0
|[1201]λ[1001]|=0
|1λ201λ|=0
(1λ)2=0
1λ=0
λ3=1
λ4=1
Thus, the matrices A and B have equal eigen values.
Step 3
Use the definition of similar matrices B=S1AS, where S is a nonsingular matrix, to check the similarity of matrices A and B.
B=S1AS
=S1I2S
=S1S
=I2
But, BI2
Hence, the matrix B is not diagonalizable, BS1AS
Therefore, the matrices A and B are not similar.
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Jeffrey Jordon
Answered 2022-01-22 Author has 2070 answers

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