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}

Harlen Pritchard 2020-12-15 Answered

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 (A1(i.e.A1=) that involves only the matrices A,BB1,C,C1,D and/or D1

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hosentak
Answered 2020-12-16 Author has 100 answers
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 A1 that involves only the matrices A,BB1,C,C1,D and/or D1
Since A, B, C and D are invertible so A1,B1,C1and D1 exists.
Given,
D=ABC
Post multiply this equation with D1 on both sides.
D(D1)=(ABC)(D1)
DD1=ABCD1      (DD1=I)
I=ABCD1    where I is the identity matrix.
Step 2
Now the equation is,
I=ABCD1
Pre multiply this equation with A1 on both sides.
(A1)I=(A1)(ABCD1)
A1I=(A1A)(BCD1)
( Matrices are associative, (AB)C=A(BC))
A1=(I)BCD1     (A1A=I)
A1=BCD1
Hence, the formula of A1 involving the matrices A,B,B1,C,C1,D and/or D1 is,
A1=BCD1
Answer: A1=BCD1
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Jeffrey Jordon
Answered 2022-01-23 Author has 2047 answers

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