15.5: Matrix Inversion
- Page ID
- 106900
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The inverse of a square matrix \(\mathbf{A}\), sometimes called a reciprocal matrix, is a matrix \(\mathbf{A}^{-1}\) such that \(\mathbf{A}\mathbf{A}^{-1}=\mathbf{I}\), where \(\mathbf{I}\) is the identity matrix.
It is easy to obtain \(\mathbf{A}^{-1}\) in the case of a \(2\times 2\) matrix:
\[\mathbf{A}=\begin{pmatrix} a&b \\ c&d \end{pmatrix};\;\mathbf{A}^{-1}=\begin{pmatrix} e&f \\ g&h \end{pmatrix} \nonumber \]
\[\begin{pmatrix} a&b \\ c&d \end{pmatrix}\begin{pmatrix} e&f \\ g&h \end{pmatrix}=\begin{pmatrix} 1&0 \\ 0&1 \end{pmatrix} \nonumber \]
\[\label{eq:matrices_inverse1} ae+bg=1 \]
\[\label{eq:matrices_inverse2} af+bh=0 \]
\[\label{eq:matrices_inverse3} ce+dg=0 \]
\[\label{eq:matrices_inverse4} cf+dh=1 \]
From Equations \ref{eq:matrices_inverse1} and \ref{eq:matrices_inverse3}: \(g=(1-ae)/b=-ce/d\rightarrow ae=cbe/d+1\rightarrow e\left(a-cb/d\right)=1\rightarrow e\left(ad-cb\right)=d\rightarrow e=d/(ad-cb)\). You can obtain expressions for \(f,g\) and \(h\) in a similar way to obtain:
\[\mathbf{A}^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d&-b \\ -c&a \end{pmatrix} \nonumber \]
Notice that the term \((ad-bc)\) is the determinant of \(\mathbf{A}\), and therefore \(\mathbf{A}^{-1}\) exists only if \(|\mathbf{A}|\neq 0\). In other words, the inverse of a singular matrix is not defined.
If you think about a square matrix as an operator, the inverse “undoes” what the original matrix does. For example, the matrix \(\begin{pmatrix} -2&0 \\ 0&1 \end{pmatrix}\), when applied to a vector \((x,y)\), gives \((-2x,y)\):
\[\begin{pmatrix} -2&0 \\ 0&1 \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}=\begin{pmatrix} -2x\\ y \end{pmatrix} \nonumber \]
The inverse of \(\mathbf{A}\), when applied to \((-2x,y)\), gives back the original vector, \((x,y)\):
\[\mathbf{A}^{-1}=\frac{1}{ad-bc}\begin{pmatrix} d&-b \\ -c&a \end{pmatrix}\rightarrow \mathbf{A}^{-1}= -\frac{1}{2}\begin{pmatrix} 1&0 \\ 0&-2 \end{pmatrix} \nonumber \]
\[-\frac{1}{2}\begin{pmatrix} 1&0 \\ 0&-2 \end{pmatrix} \begin{pmatrix} -2x\\ y \end{pmatrix}=\begin{pmatrix} x\\ y \end{pmatrix} \nonumber \]
It is of course possible to calculate the inverse of matrices of higher dimensions, but in this course you will not be required to do so by hand.