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If A and B matrices commute then...

If A and B matrices commute then

A

`A^(-1) and B ` also commute

B

` B^(-1) ` and A also commute

C

`A^(-1) and B^(-1) ` also commute

D

all the above

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to show that if matrices \( A \) and \( B \) commute, then their inverses \( A^{-1} \) and \( B^{-1} \) also commute. ### Step-by-Step Solution: 1. **Understanding the Commutative Property**: Given that matrices \( A \) and \( B \) commute, we have: \[ AB = BA \] 2. **Taking the Inverse of the Product**: We need to find the inverse of the product \( AB \). By the property of inverses, we know: \[ (AB)^{-1} = B^{-1}A^{-1} \] 3. **Using the Commutative Property**: Since \( AB = BA \), we can also express the inverse of \( BA \): \[ (BA)^{-1} = A^{-1}B^{-1} \] 4. **Equating the Inverses**: From the above two equations, we have: \[ (AB)^{-1} = B^{-1}A^{-1} \quad \text{and} \quad (BA)^{-1} = A^{-1}B^{-1} \] Since \( AB = BA \), we can equate the two: \[ B^{-1}A^{-1} = A^{-1}B^{-1} \] 5. **Conclusion**: This shows that the inverses \( A^{-1} \) and \( B^{-1} \) also commute: \[ A^{-1}B^{-1} = B^{-1}A^{-1} \] Thus, we conclude that if matrices \( A \) and \( B \) commute, then their inverses \( A^{-1} \) and \( B^{-1} \) also commute. ### Final Answer: The correct option is \( C \): If \( A \) and \( B \) commute, then \( A^{-1} \) and \( B^{-1} \) also commute.
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