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Unformatted text preview: sing row operations.
What does all of this mean for a system of linear equations? Theorem 8.6 tells us that if we write
the system in the form AX = B , then if the coeﬃcient matrix A is invertible, there is only one
solution to the system − that is, if A is invertible, the system is consistent and independent.5 We
4 If this proof sounds familiar, it should. See the discussion following Theorem 5.2 on page 295.
It can be shown that a matrix is invertible if and only if when it serves as a coeﬃcient matrix for a system of
equations, the system is always consistent independent. It amounts to the second property in Theorem 8.6 where
the matrices B are restricted to being n × 1 matrices. We note for the interested reader that, owing to how matrix
multiplication is deﬁned, being able to ﬁnd unique solutions to AX = B for n × 1 matrices B gives you the same
statement about solving such equations for n × r matrices − since we can ﬁnd a unique solution to them one column
at a time.
5 498 System...
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