Template:Matrix

The following documentation is located at Template:Matrix/doc. [<Edit> Template:Matrix/doc]
[⧼Purge⧽ Template:Matrix]

The {{matrix}} mathematical formatting template facilitates the typesetting of matrices (parenthesized matrix [default], bracketed matrix, matrix determinant, or raw matrix) for either HTML+CSS or LaTeX.

For single row or single column matrices, i.e. row vectors or column vectors, you may use the more semantically appropriate {{vector}} template, which simply redirects to the {{matrix}} template.

Usage

{{matrix|matrix entries}}

or

{{matrix|matrix entries|format}}

or

{{matrix|matrix entries|form}}

or

{{matrix|matrix entries|form|format}}

where

• matrix columns are separated by two consecutive ampersand characters &&;
• matrix rows are separated by two consecutive backslash characters \\ (do not put \\ at the end of the last row);

and form (optional second argument) is from

• ( ): parenthesized matrix (default);
• [ ]: bracketed matrix;
• det: matrix determinant;
• raw: raw matrix;

and where format (optional last argument) is from (case insensitive)

• htm: HTML+CSS (default);
• tex: LaTeX.

Examples

Raw matrix

The code

{{indent}}{{math|

'''A'''{{sub|5,4}} {{=|sp}} <!--
 
-->{{matrix|       
     ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && ''a''{{sub|1,3}} && ''a''{{sub|1,4}} && ''a''{{sub|1,5}} \\       
     ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && ''a''{{sub|2,3}} && ''a''{{sub|2,4}} && ''a''{{sub|2,5}} \\      
     ''a''{{sub|3,1}} && ''a''{{sub|3,2}} && ''a''{{sub|3,3}} && ''a''{{sub|3,4}} && ''a''{{sub|3,5}} \\       
     ''a''{{sub|4,1}} && ''a''{{sub|4,2}} && ''a''{{sub|4,3}} && ''a''{{sub|4,4}} && ''a''{{sub|4,5}}      
   |raw}}

|&&}}

yields the display style HTML+CSS raw matrix

     

A5,4  =  a1,1   a1,2   a1,3   a1,4   a1,5 a2,1   a2,2   a2,3   a2,4   a2,5 a3,1   a3,2   a3,3   a3,4   a3,5 a4,1   a4,2   a4,3   a4,4   a4,5

while the code

{{indent}}{{math| 

\mathbf{A}_{5,4} {{=}} <!--
 
-->{{matrix|
     a_{1,1} && a_{1,2} && a_{1,3} && a_{1,4} && a_{1,5} \\
     a_{2,1} && a_{2,2} && a_{2,3} && a_{2,4} && a_{2,5} \\
     a_{3,1} && a_{3,2} && a_{3,3} && a_{3,4} && a_{3,5} \\
     a_{4,1} && a_{4,2} && a_{4,3} && a_{4,4} && a_{4,5}
   |raw|tex}}

|$$}}

yields the display style LaTeX raw matrix

     ${\begin{array}{l}\displaystyle {\mathbf {A} _{5,4}={\begin{matrix}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}&a_{1,5}\\a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}&a_{2,5}\\a_{3,1}&a_{3,2}&a_{3,3}&a_{3,4}&a_{3,5}\\a_{4,1}&a_{4,2}&a_{4,3}&a_{4,4}&a_{4,5}\end{matrix}}}\end{array}}$

Parenthesized matrix

The code

{{indent}}{{math|

'''A'''{{sub|5,4}} {{=|sp}} <!--
 
-->{{matrix|       
     ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && ''a''{{sub|1,3}} && ''a''{{sub|1,4}} && ''a''{{sub|1,5}} \\       
     ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && ''a''{{sub|2,3}} && ''a''{{sub|2,4}} && ''a''{{sub|2,5}} \\      
     ''a''{{sub|3,1}} && ''a''{{sub|3,2}} && ''a''{{sub|3,3}} && ''a''{{sub|3,4}} && ''a''{{sub|3,5}} \\       
     ''a''{{sub|4,1}} && ''a''{{sub|4,2}} && ''a''{{sub|4,3}} && ''a''{{sub|4,4}} && ''a''{{sub|4,5}}      
   |( )}}

|&&}}

yields the display style HTML+CSS parenthesized matrix

     

A5,4  =  a1,1   a1,2   a1,3   a1,4   a1,5 a2,1   a2,2   a2,3   a2,4   a2,5 a3,1   a3,2   a3,3   a3,4   a3,5 a4,1   a4,2   a4,3   a4,4   a4,5

while the code

{{indent}}{{math| 

\mathbf{A}_{5,4} {{=}} <!--
 
-->{{matrix|
     a_{1,1} && a_{1,2} && a_{1,3} && a_{1,4} && a_{1,5} \\
     a_{2,1} && a_{2,2} && a_{2,3} && a_{2,4} && a_{2,5} \\
     a_{3,1} && a_{3,2} && a_{3,3} && a_{3,4} && a_{3,5} \\
     a_{4,1} && a_{4,2} && a_{4,3} && a_{4,4} && a_{4,5}
   |( )|tex}}

|$$}}

yields the display style LaTeX parenthesized matrix

     ${\begin{array}{l}\displaystyle {\mathbf {A} _{5,4}={\begin{pmatrix}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}&a_{1,5}\\a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}&a_{2,5}\\a_{3,1}&a_{3,2}&a_{3,3}&a_{3,4}&a_{3,5}\\a_{4,1}&a_{4,2}&a_{4,3}&a_{4,4}&a_{4,5}\end{pmatrix}}}\end{array}}$

The code

{{indent}}{{math|

'''A'''{{sub|''m'',''n''}} {{=|sp}} <!--
 
-->{{matrix|       
     ''a''{{sub|1,1}}      &&  ''a''{{sub|1,2}}      && {{...|cdots}}  && ''a''{{sub|1,''n''}}      \\       
     ''a''{{sub|2,1}}      &&  ''a''{{sub|2,2}}      && {{...|cdots}}  && ''a''{{sub|2,''n''}}      \\      
     {{...|vdots}}         &&  {{...|vdots}}         && {{...|ddots}}  && {{...|vdots}}             \\ 
     ''a''{{sub|''m'',1}}  &&  ''a''{{sub|''m'',2}}  && {{...|cdots}}  && ''a''{{sub|''m'',''n''}}       
   }}

|&&}} 

yields the display style HTML+CSS parenthesized matrix

     

Am,n  =  a1,1   a1,2   ⋯   a1,n a2,1   a2,2   ⋯   a2,n ⋮   ⋮   ⋱   ⋮ am,1   am,2   ⋯   am,n

while the code

{{indent}}{{math| 

\mathbf{A}_{m,n} {{=}} <!--

-->{{matrix| 
     a_{1,1}           && a_{1,2}           && {{...|cdots|tex}} & a_{1,n}            \\
     a_{2,1}           && a_{2,2}           && {{...|cdots|tex}} & a_{2,n}            \\
     {{...|vdots|tex}} && {{...|vdots|tex}} && {{...|ddots|tex}} & {{...|vdots|tex}}  \\
     a_{m,1}           && a_{m,2}           && {{...|cdots|tex}} & a_{m,n}
   |tex}}

|$$}}

yields the display style LaTeX (rendered as .PNG image) parenthesized matrix

     ${\begin{array}{l}\displaystyle {\mathbf {A} _{m,n}={\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\cdots &a_{m,n}\end{pmatrix}}}\end{array}}$

Bracketed matrix

The code

{{indent}}{{math|

'''A'''{{sub|5,4}} {{=|sp}} <!--

-->{{matrix|       
     ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && ''a''{{sub|1,3}} && ''a''{{sub|1,4}} && ''a''{{sub|1,5}} \\       
     ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && ''a''{{sub|2,3}} && ''a''{{sub|2,4}} && ''a''{{sub|2,5}} \\      
     ''a''{{sub|3,1}} && ''a''{{sub|3,2}} && ''a''{{sub|3,3}} && ''a''{{sub|3,4}} && ''a''{{sub|3,5}} \\       
     ''a''{{sub|4,1}} && ''a''{{sub|4,2}} && ''a''{{sub|4,3}} && ''a''{{sub|4,4}} && ''a''{{sub|4,5}}      
   |[ ]}}

|&&}} 

yields the display style HTML+CSS [square] bracketed matrix

     

A5,4  =  a1,1   a1,2   a1,3   a1,4   a1,5 a2,1   a2,2   a2,3   a2,4   a2,5 a3,1   a3,2   a3,3   a3,4   a3,5 a4,1   a4,2   a4,3   a4,4   a4,5

while the code

{{indent}}{{math| 

\mathbf{A}_{5,4} {{=}} <!--

-->{{matrix|       
     a_{1,1} && a_{1,2} && a_{1,3} && a_{1,4} && a_{1,5} \\
     a_{2,1} && a_{2,2} && a_{2,3} && a_{2,4} && a_{2,5} \\
     a_{3,1} && a_{3,2} && a_{3,3} && a_{3,4} && a_{3,5} \\
     a_{4,1} && a_{4,2} && a_{4,3} && a_{4,4} && a_{4,5}
   |[ ]|tex}}

|$$}}

yields the display style LaTeX (rendered as .PNG image) [square] bracketed matrix

     ${\begin{array}{l}\displaystyle {\mathbf {A} _{5,4}={\begin{bmatrix}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}&a_{1,5}\\a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}&a_{2,5}\\a_{3,1}&a_{3,2}&a_{3,3}&a_{3,4}&a_{3,5}\\a_{4,1}&a_{4,2}&a_{4,3}&a_{4,4}&a_{4,5}\end{bmatrix}}}\end{array}}$

Matrix determinant

The code

{{indent}}{{math|

det '''A'''{{sub|1,1}} {{=|sp}} <!-- 

-->{{matrix|       
     ''a''{{sub|1,1}}     
   |det}} <!-- 

-->{{=}} {{det|''a''{{sub|1,1}}}}

|&&}} 

yields the display style HTML+CSS matrix determinant (use the {{det}} template for better results with 1 × 1 matrices)

     

det A1,1  =  a1,1 = | a1,1 |

while the code

{{indent}}{{math| 

\operatorname{det} ~ \mathbf{A}_{1,1} {{=}} <!-- 

-->{{matrix|
     a_{1,1}
   |det|tex}} <!-- 

-->{{=}} {{det| a_{1,1} |tex}}

|$$}}

yields the display style LaTeX (rendered as .PNG image) matrix determinant

     ${\begin{array}{l}\displaystyle {\operatorname {det} ~\mathbf {A} _{1,1}={\begin{vmatrix}a_{1,1}\end{vmatrix}}=|a_{1,1}|}\end{array}}$

The code

{{indent}}{{math|

det '''A'''{{sub|''n'',''n''}} {{=|sp}} <!--

-->{{matrix|       
     ''a''{{sub|1,1}}      &&  ''a''{{sub|1,2}}      && {{...|cdots}}  && ''a''{{sub|1,''n''}}      \\       
     ''a''{{sub|2,1}}      &&  ''a''{{sub|2,2}}      && {{...|cdots}}  && ''a''{{sub|2,''n''}}      \\      
     {{...|vdots}}         &&  {{...|vdots}}         && {{...|ddots}}  && {{...|vdots}}             \\ 
     ''a''{{sub|''n'',1}}  &&  ''a''{{sub|''n'',2}}  && {{...|cdots}}  && ''a''{{sub|''n'',''n''}}      
   |det}}

|&&}} 

yields the display style HTML+CSS matrix determinant

     

det An,n  =  a1,1   a1,2   ⋯   a1,n a2,1   a2,2   ⋯   a2,n ⋮   ⋮   ⋱   ⋮ an,1   an,2   ⋯   an,n

while the code

{{indent}}{{math| 

\operatorname{det} ~ \mathbf{A}_{n,n} {{=}} <!--

-->{{matrix|
     a_{1,1} && a_{1,2} && \cdots && a_{1,n} \\
     a_{2,1} && a_{2,2} && \cdots && a_{2,n} \\
     \vdots  && \vdots  && \ddots && \vdots  \\
     a_{n,1} && a_{n,2} && \cdots && a_{n,n}
   |det|tex}}

|$$}}

yields the display style LaTeX (rendered as .PNG image) matrix determinant

     ${\begin{array}{l}\displaystyle {\operatorname {det} ~\mathbf {A} _{n,n}={\begin{vmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,n}\end{vmatrix}}}\end{array}}$

Matrix inverse formula

The code

{{indent}}{{math|<!-- 

-->'''A'''{{^|{{sp|-3}}{{op|-}}1}} {{=|sp}} {{frac|'''C'''{{sp|1}}{{sup|T}}|{{det|'''A'''}}}} {{=|sp}} <!-- 

-->{{frac|
     {{matrix|       
       ''C''{{sub|1,1}}      &&  ''C''{{sub|1,2}}      && {{...|cdots}}  && ''C''{{sub|1,''n''}}      \\       
       ''C''{{sub|2,1}}      &&  ''C''{{sub|2,2}}      && {{...|cdots}}  && ''C''{{sub|2,''n''}}      \\      
       {{...|vdots}}         &&  {{...|vdots}}         && {{...|ddots}}  && {{...|vdots}}             \\ 
       ''C''{{sub|''n'',1}}  &&  ''C''{{sub|''n'',2}}  && {{...|cdots}}  && ''C''{{sub|''n'',''n''}}      
     }} {{sup|{{sup|{{sup|{{sup|{{sup|{{sup|{{sup|T|100%}}|100%}}|100%}}|100%}}|100%}}|100%}}}}<!-- Yes, kludgy! I'll work out something better... -->
   | 
     {{matrix|       
       ''a''{{sub|1,1}}      &&  ''a''{{sub|1,2}}      && {{...|cdots}}  && ''a''{{sub|1,''n''}}      \\       
       ''a''{{sub|2,1}}      &&  ''a''{{sub|2,2}}      && {{...|cdots}}  && ''a''{{sub|2,''n''}}      \\      
       {{...|vdots}}         &&  {{...|vdots}}         && {{...|ddots}}  && {{...|vdots}}             \\ 
       ''a''{{sub|''n'',1}}  &&  ''a''{{sub|''n'',2}}  && {{...|cdots}}  && ''a''{{sub|''n'',''n''}}      
     |det}}
   |HTM}},

|&&}}{{nl}}

where {{math|{{det|'''A'''}}|&}} is the [[determinant]] of {{math|'''A'''|&}}, {{math|'''C'''|&}} is the [[matrix of cofactors]] of <!--
-->{{math|'''A'''|&}}, and {{math|'''C'''{{sup|T}}|&}} represents the matrix [[transpose]] of {{math|'''C'''|&}}. 

yields the display style HTML+CSS matrix inverse formula (using {{^}} or {{sup}} after large delimiters, e.g. the  

T

for matrix transpose, has to be raised, either manually or [hopefully] automatically)
     

A  − 1  =  C T | A |  =  C1,1   C1,2   ⋯   C1,n C2,1   C2,2   ⋯   C2,n ⋮   ⋮   ⋱   ⋮ Cn,1   Cn,2   ⋯   Cn,n T a1,1   a1,2   ⋯   a1,n a2,1   a2,2   ⋯   a2,n ⋮   ⋮   ⋱   ⋮ an,1   an,2   ⋯   an,n ,

where 

| A |

is the determinant of 

A

, 

C

is the matrix of cofactors of 

A

, and 

CT

represents the matrix transpose of 

C

.

The code

{{indent}}{{math|<!-- 

-->\mathbf{A}{{^|\! {{op|-}}1|tex}}  {{=}} {{frac|\mathbf{C}{{sup|{\rm T}|tex}}|{{det|\mathbf{A}|tex}}|tex}} {{=}}  <!--

-->{{frac|
     {{matrix| 
       ''C''{{sub|1,1|tex}}      &&  ''C''{{sub|1,2|tex}}      && {{...|cdots|tex}}  && ''C''{{sub|1,''n''|tex}}      \\       
       ''C''{{sub|2,1|tex}}      &&  ''C''{{sub|2,2|tex}}      && {{...|cdots|tex}}  && ''C''{{sub|2,''n''|tex}}      \\      
       {{...|vdots|tex}}         &&  {{...|vdots|tex}}         && {{...|ddots|tex}}  && {{...|vdots|tex}}             \\ 
       ''C''{{sub|''n'',1|tex}}  &&  ''C''{{sub|''n'',2|tex}}  && {{...|cdots|tex}}  && ''C''{{sub|''n'',''n''|tex}}       
     ||tex}}{{sup|{\rm T}|tex}}
   |
     {{matrix| 
       ''a''{{sub|1,1|tex}}      &&  ''a''{{sub|1,2|tex}}      && {{...|cdots|tex}}  && ''a''{{sub|1,''n''|tex}}      \\       
       ''a''{{sub|2,1|tex}}      &&  ''a''{{sub|2,2|tex}}      && {{...|cdots|tex}}  && ''a''{{sub|2,''n''|tex}}      \\      
       {{...|vdots|tex}}         &&  {{...|vdots|tex}}         && {{...|ddots|tex}}  && {{...|vdots|tex}}             \\ 
       ''a''{{sub|''n'',1|tex}}  &&  ''a''{{sub|''n'',2|tex}}  && {{...|cdots|tex}}  && ''a''{{sub|''n'',''n''|tex}}       
     |det|tex}}
   |TEX}},

|$$}}{{nl}}

where {{math|{{det|\mathbf{A}|tex}}|$}} is the [[determinant]] of {{math|\mathbf{A}|$}}, {{math|\mathbf{C}|$}} is the [[matrix of cofactors]] of <!--
-->{{math|\mathbf{A}|$}}, and {{math|\mathbf{C}{{sup|{\rm T}|tex}}|$}} represents the matrix [[transpose]] of {{math|\mathbf{C}|$}}.

yields the display style LaTeX matrix inverse formula

     ${\begin{array}{l}\displaystyle {\mathbf {A} ^{\!-1}={\frac {\mathbf {C} ^{\rm {T}}{\!\,\!}}{|\mathbf {A} |}}={\frac {{\begin{pmatrix}C_{1,1}{\!\,\!}&C_{1,2}{\!\,\!}&\cdots &C_{1,n}{\!\,\!}\\C_{2,1}{\!\,\!}&C_{2,2}{\!\,\!}&\cdots &C_{2,n}{\!\,\!}\\\vdots &\vdots &\ddots &\vdots \\C_{n,1}{\!\,\!}&C_{n,2}{\!\,\!}&\cdots &C_{n,n}{\!\,\!}\end{pmatrix}}^{\rm {T}}{\!\,\!}}{\begin{vmatrix}a_{1,1}{\!\,\!}&a_{1,2}{\!\,\!}&\cdots &a_{1,n}{\!\,\!}\\a_{2,1}{\!\,\!}&a_{2,2}{\!\,\!}&\cdots &a_{2,n}{\!\,\!}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}{\!\,\!}&a_{n,2}{\!\,\!}&\cdots &a_{n,n}{\!\,\!}\end{vmatrix}}},}\end{array}}$

where$\textstyle {|\mathbf {A} |}$ is the determinant of$\textstyle {\mathbf {A} }$,$\textstyle {\mathbf {C} }$ is the matrix of cofactors of$\textstyle {\mathbf {A} }$, and$\textstyle {\mathbf {C} ^{\rm {T}}{\!\,\!}}$ represents the matrix transpose of$\textstyle {\mathbf {C} }$.

Nested matrices

The code

{{indent}}{{math|

{{matrix|

{{matrix|       
  ''a'' && ''b'' \\       
  ''c'' && ''d''           
}}

&&

{{matrix|       
  ''e'' && ''f'' \\       
  ''g'' && ''h''           
}}

\\

0

&&

{{matrix|       
  ''i'' && ''j'' \\       
  ''k'' && ''l''           
}}

}}

|&&}}

yields the display style HTML+CSS

     

a   b c   d   e   f g   h 0   i   j k   l

The code

{{indent}}{{math|

{{matrix|

{{matrix|       
  ''a'' && ''b'' \\       
  ''c'' && ''d''           
|tex}}

&&

{{matrix|       
  ''e'' && ''f'' \\       
  ''g'' && ''h''           
|tex}}

\\

0

&&

{{matrix|       
  ''i'' && ''j'' \\       
  ''k'' && ''l''           
|tex}}

|tex}}

|$$}}

yields the display style LaTeX

     ${\begin{array}{l}\displaystyle {\begin{pmatrix}{\begin{pmatrix}a&b\\c&d\end{pmatrix}}&{\begin{pmatrix}e&f\\g&h\end{pmatrix}}\\0&{\begin{pmatrix}i&j\\k&l\end{pmatrix}}\end{pmatrix}}\end{array}}$

See also

• {{det}} (matrix determinant template)