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Question from Marsia, a student:

explain how inverses are used to solve linear systems.

Marsia,

Think about solving a simple algebraic equation such as

5x = 10

I expect that you would divide both sides by 5 to obtain

x = 2

But 1/5 is 5-1 which is the inverse of 5 so you can think about solving

5x = 10

as multiplying both sides by the inverse of 5 to obtain

5x = 10 thus

5-1 × 5 x = 5-1 × 10

but 5-1 × 5 = 1 so

1 × x = 5-1 × 10 or
x = 5-1 × 10

The same technique works with a linear system if you can write it as

A X = B

and A has an inverse A-1. Then A-1 ×A = I and hence

A X = B thus

A-1 × A × X = A-1 × B so

I × X = A-1 × B or

X = A-1 × B

Penny

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