To verify that matrix $A = \begin{bmatrix} -4 & -1 \\ 4 & 0 \end{bmatrix}$ is similar to itself, and that it is similar to the matrices $B$ and $C$, we will use similarity transformations as follows:

Step 1: Verify that $A$ is similar to itself

Since similarity is reflexive, $A$ should be similar to itself with an identity transformation. For $T$ such that $A = T^{-1} A T$, we can take $T$ to be the identity matrix: $T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ Thus, $T^{-1} A T = A$, which shows that $A$ is similar to itself.

Step 2: Verify that $B \sim A$

We know that $A = P^{-1} B P$ for $P = \begin{bmatrix} 2 & 1 \\ -3 & -1 \end{bmatrix}$ We need to find a matrix $S$ such that $B = S^{-1} A S$.

To find $S$, we can use the fact that $S = P$, as: $S = \begin{bmatrix} 2 & 1 \\ -3 & -1 \end{bmatrix}$ Thus, $B = S^{-1} A S$ which confirms that $B$ is similar to $A$.

Step 3: Verify that $A \sim C$

Given that $B = Q^{-1} C Q$ where $Q = \begin{bmatrix} -2 & -1 \\ 1 & 1 \end{bmatrix}$ we need to find a matrix $R$ such that $A = R^{-1} C R$.

Since we already know that $A = P^{-1} B P$ and $B = Q^{-1} C Q$, we can set up $R = P Q$. Calculating $R$: $R = P Q = \begin{bmatrix} 2 & 1 \\ -3 & -1 \end{bmatrix} \begin{bmatrix} -2 & -1 \\ 1 & 1 \end{bmatrix}$ Calculating each element of $R$:

1. First row, first column: $2(-2) + 1(1) = -4 + 1 = -3$
2. First row, second column: $2(-1) + 1(1) = -2 + 1 = -1$
3. Second row, first column: $-3(-2) + (-1)(1) = 6 - 1 = 5$
4. Second row, second column: $-3(-1) + (-1)(1) = 3 - 1 = 2$

So we have: $R = \begin{bmatrix} -3 & -1 \\ 5 & 2 \end{bmatrix}$ This gives us the matrix $R$ such that $A = R^{-1} C R$.

Summary of Results

• Matrix $T$: $T = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
• Matrix $S$: $S = \begin{bmatrix} 2 & 1 \\ -3 & -1 \end{bmatrix}$
• Matrix $R$: $R = \begin{bmatrix} -3 & -1 \\ 5 & 2 \end{bmatrix}$

These matrices verify the similarity relations among $A$, $B$, and $C$.

Do you need further details or explanations on any step?

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Here are five related questions:

1. What are the properties of similarity transformations in matrix algebra?
2. How can we determine if two matrices are similar without computing explicit transformations?
3. How does similarity affect the eigenvalues and eigenvectors of matrices?
4. Can similarity be used to simplify matrix computations in certain cases?
5. How do similarity relations relate to the concept of matrix diagonalization?

Tip: When verifying matrix similarity, always check if the inverse of the transformation matrix exists, as similarity requires invertibility.