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=⋯) that involves only the matrices A,BB−1,C,C−1,D and/or D−1

Question

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=⋯) that involves only the matrices A,BB−1,C,C−1,D and/or D−1

hosentak · Accepted Answer

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 A,BB−1,C,C−1,D and/or D−1  Since A, B, C and D are invertible so A−1,B−1,C−1and D−1 exists.  Given,  D=ABC  Post multiply this equation with D−1 on both sides.  D(D−1)=(ABC)(D−1)  DD−1=ABCD−1      (DD−1=I)  I=ABCD−1    where I is the identity matrix.  Step 2  Now the equation is,  I=ABCD−1  Pre multiply this equation with A−1  on both sides.  (A−1)I=(A−1)(ABCD−1)  A−1I=(A−1A)(BCD−1)  (∵ Matrices are associative, (AB)C=A(BC))  A−1=(I)BCD−1     (A−1A=I)  A−1=BCD−1  Hence, the formula of A−1 involving the matrices A,B,B−1,C,C−1,D and/or D−1 is,  A−1=BCD−1  Answer: A−1=BCD−1

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

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}

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