Skip to main content Contents Prev Up Next \(\newcommand{\avec}{{\mathbf a}}
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Chapter 9 Invertibility
Up to this point, we have used the Gaussian elimination algorithm to find solutions to linear systems. We now investigate another way to find solutions to the equation \(A\xvec=\bvec\) when the matrix \(A\) has the same number of rows and columns. To get started, let’s look at some familiar examples.
Explain how you would solve the equation \(3x =
5\) using multiplication rather than division.
One possible response is to divide both sides by 3. Instead, let’s rephrase this as multiplying by \(3^{-1} = \frac
13\text{,}\) the multiplicative inverse of 3.
Now that we are interested in solving equations of the form \(A\xvec = \bvec\text{,}\) we might try to find a similar approach. Is there a matrix \(A^{-1}\) that plays the role of the multiplicative inverse of \(A\text{?}\)
