66.17: Matrix Properties
- Page ID
- 91947
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)This appendix presents a brief summary of the properties of \(2 \times 2\) and \(3 \times 3\) matrices.
\(2 \times 2\) Matrices
Determinant
The determinant of a \(2 \times 2\) matrix is given by the well-known formula:
\[
\operatorname{det}\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)=a d-b c \text {. }
\]
Matrix of Cofactors
The matrix of cofactors is the matrix of signed minors; for a \(2 \times 2\) matrix, this is
\[
\operatorname{cof}\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)=\left(\begin{array}{cc}
d & -c \\
-b & a
\end{array}\right)
\]
Inverse
Finally, the inverse of a matrix is the transpose of the matrix of cofactors divided by the determinant. For a \(2 \times 2\) matrix,
\[
\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right)^{-1}=\frac{1}{a d-b c}\left(\begin{array}{cc}
d & -b \\
-c & a
\end{array}\right)
\]
\(3 \times 3\) Matrices
Determinant
The determinant of a \(3 \times 3\) matrix is given by:
\[
\operatorname{det}\left(\begin{array}{lll}
a & b & c \\
d & e & f \\
g & h & i
\end{array}\right)=a(e i-f h)-b(d i-f g)+c(d h-e g)
\]
Matrix of Cofactors
The matrix of cofactors is the matrix of signed minors; for a \(3 \times 3\) matrix, this is
\[
\operatorname{cof}\left(\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array}\right)=\left(\begin{array}{ccc}
e i-f h & f g-d i & d h-e g \\
c h-b i & a i-c g & b g-a h \\
b f-c e & c d-a f & a e-b d
\end{array}\right)
\]
Inverse
Finally, the inverse of a matrix is the transpose of the matrix of cofactors divided by the determinant. For a \(3 \times 3\) matrix,
\[
\left(\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array}\right)^{-1}=\frac{1}{a(e i-f h)-b(d i-f g)+c(d h-e g)}\left(\begin{array}{ccc}
e i-f h & c h-b i & b f-c e \\
f g-d i & a i-c g & c d-a f \\
d h-e g & b g-a h & a e-b d
\end{array}\right)
\]