4.2. Garden¶
This page is a collection of a variety of matrices used in different examples and contexts and their associated results.
4.2.1. Square Matrices¶
4.2.1.1. 2x2 matrices¶
Simple examples
\[\begin{split}A &= \begin{bmatrix}
3 & 1 \\ 5 & 2
\end{bmatrix}\\
det(A) &= 1\\
A^{-1} &= \begin{bmatrix}2 & -1\\ -5 & 3 \end{bmatrix}\end{split}\]
Next
\[\begin{split}A &= \begin{bmatrix}3&2\\4&0\end{bmatrix}\\
det(A) &= -8\\
A^{-1} &= \begin{bmatrix}0 & 1/4 \\ 1/2 & -3/8 \end{bmatrix}\end{split}\]
4.2.1.2. 3x3 matrices¶
\[\begin{split}A &= \begin{bmatrix}
1 & 3 & 2\\
4 & 1 & 3\\
2 & 5 & 2
\end{bmatrix}\\
minor(A) &= \begin{bmatrix}
-13 & 2 & 18 \\
-4 & -2 & -1 \\
7 & -5 & -11
\end{bmatrix}\\
cof(A) &= \begin{bmatrix}
-13 & -2 & 18 \\
4 & -2 & 1 \\
7 & 5 & -11
\end{bmatrix}\\
adj(A) &= \begin{bmatrix}
-13 & 4 & 7 \\
-2 & -2 & 5\\
18 & 1 & -11
\end{bmatrix}\\
det(A) &= 17\\
A^{-1} &= \begin{bmatrix}
-13/17 & 4/17 & 7/17 \\
-2/17 & -2/17 & 5/17\\
18/17 & 1/17 & -11/17
\end{bmatrix}\end{split}\]
4.2.1.3. 4x4 matrices¶
\[\begin{split}S &= \begin{bmatrix}
3 & 2 & 0 & 1\\
4 & 0 & 1 & 2 \\
3 & 0 & 2 & 1\\
9 & 2 & 3 & 1
\end{bmatrix}\\
|S| &= 24\end{split}\]
Lets write this as
\[\begin{split}S &= \begin{bmatrix} A & B \\ C & D \end{bmatrix}\end{split}\]
Where
\[\begin{split}A &= \begin{bmatrix} 3 & 2 \\ 4 & 0 \end{bmatrix}\\
B &= \begin{bmatrix} 0 & 1 \\ 1 & 2 \end{bmatrix}\\
C &= \begin{bmatrix} 3 & 0 \\ 9 & 2 \end{bmatrix}\\
D &= \begin{bmatrix} 2 & 1 \\ 3 & 1 \end{bmatrix}\end{split}\]
We have:
\[|S| = |A| |D - CA^{-1}B|\]
\[\begin{split}|A| &= -8\\
A^{-1} &= \begin{bmatrix}0 & 1/4 \\ 1/2 & -3/8 \end{bmatrix}\\
CA^{-1} &= \begin{bmatrix} 0 & 3/4 \\ 1 & 3/2 \end{bmatrix}\\
CA^{-1}B &= \begin{bmatrix} 3/4 & 3/2 \\ 3/2& 4\end{bmatrix}\\
D - CA^{-1}B &= \begin{bmatrix} 5/4 & -1/2 \\ 3/2& -3\end{bmatrix}\\
|D - CA^{-1}B| &= -3\\
|S| &= |A||D - CA^{-1}B| = -8 \times -3 = 24\end{split}\]
Inverse
\[\begin{split}\begin{bmatrix} A & B \\ C & D \end{bmatrix}
\begin{bmatrix} E & F \\ G & H \end{bmatrix} = I\end{split}\]
gives
\[\begin{split}E &= (A - BD^{-1}C)^{-1}\\
G &= -D^{-1}CE\\
H &= (D - CA^{-1}B)^{-1}\\
F &= -A^{-1}BH\end{split}\]
Algorithm:
\(P = D^{-1}C\)
\(E = (A - BP)^{-1}\)
\(G = - P E\)
\(Q = A^{-1}B\)
\(H = (D - CQ)^{-1}\)
\(F=-Q H\)
4.2.1.4. Special 2x2 matrices¶
Following are 2x2 matrices such that \(A^2 = A\).
\[ \begin{align}\begin{aligned}\begin{split}\left[\begin{array}{cc}1 & 0 \\ 0 & 0\end{array}\right]\end{split}\\\begin{split}\left[\begin{array}{cc}0 & 0 \\ 0 & 1\end{array}\right]\end{split}\\\begin{split}\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\end{split}\\\begin{split}\left[\begin{array}{cc}1 & 1 \\ 0 & 0\end{array}\right]\end{split}\\\begin{split}\left[\begin{array}{cc}0 & 0 \\ 1 & 1\end{array}\right]\end{split}\\\begin{split}\left[\begin{array}{cc}1 & 0 \\ 1 & 0\end{array}\right]\end{split}\\\begin{split}\left[\begin{array}{cc}0 & 1 \\ 0 & 1\end{array}\right]\end{split}\end{aligned}\end{align} \]
Flip matrix
\[\begin{split}\left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right]
\left[\begin{array}{cc}0 & 1 \\ 1 & 0\end{array}\right]
=
\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]\end{split}\]