Differential Equations and Linear Algebra, 4th Edition

3.5 Inverses of Matrices

Recall that the $n\times n$ identity matrix is the diagonal matrix

$$\mathbf{\text{I}}=\left[\begin{array}{ccccc}1& 0& 0& \cdots & 0\\ 0& 1& 0& \cdots & 0\\ 0& 0& 1& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & 1\end{array}\right]$$ (1)

having ones on its main diagonal and zeros elsewhere. It is not difficult to deduce directly from the definition of the matrix product that I acts like an identity for matrix multiplication:

$$\mathbf{\text{AI}}=\mathbf{\text{A}}\text{and}\mathbf{\text{IB}}=\mathbf{\text{B}}$$ (2)

if the sizes of A and B are such that the products AI and IB are defined. It is, nevertheless, instructive to derive the identities in (2) formally from the two basic facts about matrix multiplication that we state below. First, recall that the notation

$$\mathbf{\text{A}}=\left[\begin{array}{lllll}{\mathbf{\text{a}}}_1& {\mathbf{\text{a}}}_2& {\mathbf{\text{a}}}_3& \cdots & {\mathbf{\text{a}}}_n\end{array}\right]$$ (3)

expresses the $m\times n$ matrix A in terms of its column vectors ${\mathbf{\text{a}}}_1,{\mathbf{\text{a}}}_2,{\mathbf{\text{a}}}_3,\dots ,{\mathbf{\text{a}}}_n$.

1. Fact 1 Ax in terms of columns of A

   If $\mathbf{\text{A}}=\left[\begin{array}{llll}{\mathbf{\text{a}}}_1& {\mathbf{\text{a}}}_2& \cdots & {\mathbf{\text{a}}}_n\end{array}\right]$ and $\mathbf{\text{x}}=(x_1,x_2,\dots ,x_n)$ is an n-vector, ...