# If the product D=ABC of three square matrices is invertible , then A must be invertible (so are B and C). Find a formula for A^{-1} (i.e. A^{-1}=dotsb) that involves only the matrices A, B B^{-1} , C, C^{-1} , D text{ and/or } D^{-1}

If the product D=ABC of three square matrices is invertible , then A must be invertible (so are B and C). Find a formula for $\left({A}^{-1}\left(i.e.{A}^{-1}=\cdots \right)$ that involves only the matrices

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Step 1
Given that the product D=ABC of three square matrices is invertible.
Also given that A must be invertible and so are B and C.
To find ${A}^{-1}$ that involves only the matrices
Since A, B, C and D are invertible so exists.
Given,
D=ABC
Post multiply this equation with ${D}^{-1}$ on both sides.
$D\left({D}^{-1}\right)=\left(ABC\right)\left({D}^{-1}\right)$

Step 2
Now the equation is,
$I=ABC{D}^{-1}$
Pre multiply this equation with ${A}^{-1}$ on both sides.
$\left({A}^{-1}\right)I=\left({A}^{-1}\right)\left(ABC{D}^{-1}\right)$
${A}^{-1}I=\left({A}^{-1}A\right)\left(BC{D}^{-1}\right)$

${A}^{-1}=BC{D}^{-1}$
Hence, the formula of ${A}^{-1}$ involving the matrices is,
${A}^{-1}=BC{D}^{-1}$
Answer: ${A}^{-1}=BC{D}^{-1}$
Jeffrey Jordon