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If A, B, C are invertible matrices, then...

If A, B, C are invertible matrices, then `(ABC)^(-1)`=

A

`A^(-1) B^(-1) C^(-1) `

B

` B^(-1) C^(-1) A^(-1) `

C

` C^(-1) B^(-1) A^(-1) `

D

can not said

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The correct Answer is:
To find the inverse of the product of three invertible matrices \( A \), \( B \), and \( C \), we can use the property of inverses which states that the inverse of a product of matrices is equal to the product of their inverses taken in reverse order. ### Step-by-Step Solution: 1. **Start with the expression**: We want to find \( (ABC)^{-1} \). 2. **Apply the property of inverses**: According to the property of inverses, we have: \[ (XY)^{-1} = Y^{-1}X^{-1} \] Therefore, for three matrices, we can extend this property: \[ (ABC)^{-1} = C^{-1}B^{-1}A^{-1} \] 3. **Write the final result**: Thus, we can conclude that: \[ (ABC)^{-1} = C^{-1}B^{-1}A^{-1} \] ### Final Answer: The inverse of the product of matrices \( A \), \( B \), and \( C \) is: \[ (ABC)^{-1} = C^{-1}B^{-1}A^{-1} \]
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