Let A=[1001] and B=[1201] , show that A and B are not similar.

Step 1 To check the similarity of the matrices, calculate the determinant of the matrices A=[1001] and B=[1201] |A|=|1001| =1−0 =1 |B|=|1201| =1−0 =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. |A−λI|=0 |[1001]−λ[1001]|=0 |1−λ001−λ|=0 (1−λ)2=0 1−λ=0 λ1=1 λ2=1 |B−λI|=0 |[1201]−λ[1001]|=0 |1−λ201−λ|=0 (1−λ)2=0 1−λ=0 λ3=1 λ4=1 Thus, the matrices A and B have equal eigen values. Step 3 Use the definition of similar matrices B=S−1AS, where S is a nonsingular matrix, to check the similarity of matrices A and B. B=S−1AS =S−1I2S =S−1S =I2 But, B≠I2 Hence, the matrix B is not diagonalizable, B≠S−1AS Therefore, the matrices A and B are not similar.

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tabita57i

Answered question

2020-11-08

Let

A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]

, show that A and B are not similar.

Answer & Explanation

gotovub

Skilled2020-11-09Added 98 answers

Step 1
To check the similarity of the matrices, calculate the determinant of the matrices

A = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]

| \begin{matrix} A \end{matrix} | = | \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} |

= 1 - 0

= 1

| \begin{matrix} B \end{matrix} | = | \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix} |

= 1 - 0

= 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.

| \begin{matrix} A - λ I \end{matrix} | = 0

| \begin{matrix} [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] - λ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{matrix} | = 0

| \begin{matrix} 1 - λ & 0 \\ 0 & 1 - λ \end{matrix} | = 0

(1 - λ)^{2} = 0

1 - λ = 0

λ_{1} = 1

λ_{2} = 1

| \begin{matrix} B - λ I \end{matrix} | = 0

| \begin{matrix} [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}] - λ [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] \end{matrix} | = 0

| \begin{matrix} 1 - λ & 2 \\ 0 & 1 - λ \end{matrix} | = 0

(1 - λ)^{2} = 0

1 - λ = 0

λ_{3} = 1

λ_{4} = 1

Thus, the matrices A and B have equal eigen values.
Step 3
Use the definition of similar matrices

B = S^{- 1} A S

, where S is a nonsingular matrix, to check the similarity of matrices A and B.

B = S^{- 1} A S

= S^{- 1} I_{2} S

= S^{- 1} S

= I_{2}

But, B \neq I_{2}

Hence, the matrix B is not diagonalizable,

B \neq S^{- 1} A S

Therefore, the matrices A and B are not similar.

Jeffrey Jordon

Expert2022-01-22Added 2605 answers

Answer is given below (on video)

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