66.17: Matrix Properties
- Page ID
- 91947
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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,
\[
\left(\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array}\right)^{-1}=\frac{1}{a(e i-f h)-b(d i-f g)+c(d h-e g)}\left(\begin{array}{ccc}
e i-f h & c h-b i & b f-c e \\
f g-d i & a i-c g & c d-a f \\
d h-e g & b g-a h & a e-b d
\end{array}\right)
\]