## Matrix

martix parentheses

$\begin{pmatrix} a & b \\ c & d \end{pmatrix}$

\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}

If you want to enclose the matrix in parentheses, you can use the "pmatrix" (p from parentheses +matrix). If you use just "matrix" without adding "p," the matrix will be displayed without parentheses.

martix brackets

$\begin{bmatrix} a & b \\ c & d \end{bmatrix}$

\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}

To enclose a matrix in brackets, you can use the "bmatrix" (b from brackets +matrix). However, if you want the brackets to be curly brackets, you can use the "Bmatrix", with a capital "B".

martix vertical lines

$\begin{vmatrix} a & b \\ c & d \end{vmatrix}$

\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}

To enclose a matrix in vertical lines, you can use the "vmatrix" (v from vertical lines +matrix). However, if you want the vertical lines to be double lines, you can use the "Vmatrix", with a capital "V".

transposed matrix

$A^{ \mathrm{ T } }$

A^{ \mathrm{ T } }

To represent the transpose of a matrix, you can use a Roman font uppercase "T" in the top right corner of the matrix.

transposed matrix 2

${}^t \! A$

{}^t \! A

It is also common to represent the transpose of a matrix by writing a lowercase "t" in the top left corner of the matrix.

dimension

$\dim$

\dim

The notation "dimension" is derived from the dimension.

matrix rank

$\mathrm{ rank } A$

\mathrm{ rank } A
trace

$\mathrm{ Tr } A$

\mathrm{ Tr } A
determinant

$\mathrm{ det }A$

\mathrm{ det }A
determinant sample

$\begin{eqnarray} \mathrm{ det }A = | A | = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \end{eqnarray}$

\begin{eqnarray}
\mathrm{ det }A
= | A |
= \begin{vmatrix} a & b \\ c & d \end{vmatrix}
\end{eqnarray}
martix large

$\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & I \end{pmatrix}$

\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & I
\end{pmatrix}

Using the "pmatrix" environment, you can write a 3x3 matrix.

martix large

$\begin{eqnarray} \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
\left(
\begin{array}{ccc}
a & b & c \\
d & e & f \\
g & h & i
\end{array}
\right)
\end{eqnarray}

Using the "array" environment, you can write a matrix. The "ccc" is for center alignment.

right alignment matrix

$\begin{eqnarray} \left( \begin{array}{rrr} 111 & 111 & 111 \\ 22 & 0.2 & -2 \\ 3 & 3 & 3 \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
\left(
\begin{array}{rrr}
111 & 111 & 111 \\
22 & 0.2 & -2 \\
3 & 3 & 3
\end{array}
\right)
\end{eqnarray}

Using the "array" environment, you can write a matrix with right alignment for each value by specifying "rrr".

mxn matrix

$\begin{eqnarray} A = \left( \begin{array}{cccc} a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\ a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\ \vdots & \vdots & \ddots & \vdots \\ a_{ m1 } & a_{ m2 } & \ldots & a_{ mn } \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
A = \left(
\begin{array}{cccc}
a_{ 11 } & a_{ 12 } & \ldots & a_{ 1n } \\
a_{ 21 } & a_{ 22 } & \ldots & a_{ 2n } \\
\vdots & \vdots & \ddots & \vdots \\
a_{ m1 } & a_{ m2 } & \ldots & a_{ mn }
\end{array}
\right)
\end{eqnarray}

It is an example of representing an m x n matrix using multiple dots.

block matrix

$\begin{eqnarray} \left( \begin{array}{cc|cc} a & b & 0 & 0 \\ c & d & 0 & 0 \\ \hline x & y & 1 & 0 \\ z & w & 0 & 1 \\ \end{array} \right) \end{eqnarray}$

\begin{eqnarray}
\left(
\begin{array}{cc|cc}
a & b & 0 & 0 \\
c & d & 0 & 0 \\
\hline
x & y & 1 & 0 \\
z & w & 0 & 1 \\
\end{array}
\right)
\end{eqnarray}

Using "|" like the "cc|cc", you can draw vertical lines between columns. The "\hline" command can be used to draw horizontal lines.

Jordan block

$\begin{eqnarray} \begin{pmatrix} \lambda & 1 & & & 0 \\ & \lambda & 1 & & \\ & & \ddots & \ddots & \\ & & & \lambda & 1 \\ 0 & & & & \lambda \end{pmatrix} \end{eqnarray}$

\begin{eqnarray}
\begin{pmatrix}
\lambda & 1 &   &  & 0 \\
& \lambda & 1 &   &   \\
&   & \ddots & \ddots &   \\
&   &   & \lambda & 1  \\
0 &   &   &   & \lambda
\end{pmatrix}
\end{eqnarray}
cofactor

$\begin{eqnarray} & & (-1)^{ i+j } \times \\[5pt] & & \quad \begin{vmatrix} a_{1,1} & \ldots & a_{1,j-1} & a_{1,j+1} & \ldots & a_{1,n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{i-1,1} & \ldots & a_{i-1, j-1} & a_{i-1, j+1} & \ldots & a_{i-1, n} \\ a_{i+1,1} & \ldots & a_{i+1, j-1} & a_{i+1, j+1} & \ldots & a_{i+1, n} \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & \ldots & a_{n, j-1} & a_{n, j+1} & \ldots & a_{n, n} \end{vmatrix} \end{eqnarray}$

\begin{eqnarray}
& & (-1)^{ i+j } \times \\[5pt]
\end{eqnarray}