If A , B and C are three square matrices of the same size such that B = `CA C^(-1)` then `CA^(3) C^(-1)` is equal to

Options

A. B
B. `B^(2)`
C. `B^(3)`
D. `B^(9)`

Answer

To solve the problem, we need to find the expression \( CA^3C^{-1} \) given that \( B = CAC^{-1} \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ B = CAC^{-1} \]

2. **Multiply both sides by \( B \):**
\[ B^2 = B \cdot B = (CAC^{-1})(CAC^{-1}) \]

3. **Substitute \( B \) in the right-hand side:**
\[ B^2 = CAC^{-1}CAC^{-1} \]

4. **Rearranging the right-hand side:**
\[ B^2 = CA(C^{-1}C)AC^{-1} \]
Since \( C^{-1}C \) is the identity matrix \( I \):
\[ B^2 = CAIA C^{-1} = CA^2C^{-1} \]

5. **Now we have:**
\[ B^2 = CA^2C^{-1} \]

6. **Multiply both sides by \( B \) again:**
\[ B^3 = B \cdot B^2 = (CAC^{-1})(CA^2C^{-1}) \]

7. **Substituting \( B^2 \):**
\[ B^3 = CAC^{-1}CA^2C^{-1} \]

8. **Rearranging the right-hand side:**
\[ B^3 = CA(C^{-1}C)A^2C^{-1} \]
Again, \( C^{-1}C = I \):
\[ B^3 = CAIA^2C^{-1} = CA^3C^{-1} \]

9. **Thus, we conclude:**
\[ CA^3C^{-1} = B^3 \]

### Final Answer:
\[ CA^3C^{-1} = B^3 \]