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# 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}}

|$$}}  yields the display style LaTeX raw matrix ${\displaystyle {\begin{array}{l}\displaystyle {\mathbf {A} _{5,4}={\begin{matrix}a_{1,1}&a_{1,2}&a_{1,3}&a_{1,4}&a_{1,5}\\a_{2,1}&a_{2,2}&a_{2,3}&a_{2,4}&a_{2,5}\\a_{3,1}&a_{3,2}&a_{3,3}&a_{3,4}&a_{3,5}\\a_{4,1}&a_{4,2}&a_{4,3}&a_{4,4}&a_{4,5}\end{matrix}}}\end{array}}}$ ### Parenthesized matrix The code {{indent}}{{math| '''A'''{{sub|5,4}} {{=|sp}} <!-- -->{{matrix| ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && ''a''{{sub|1,3}} && ''a''{{sub|1,4}} && ''a''{{sub|1,5}} \\ ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && ''a''{{sub|2,3}} && ''a''{{sub|2,4}} && ''a''{{sub|2,5}} \\ ''a''{{sub|3,1}} && ''a''{{sub|3,2}} && ''a''{{sub|3,3}} && ''a''{{sub|3,4}} && ''a''{{sub|3,5}} \\ ''a''{{sub|4,1}} && ''a''{{sub|4,2}} && ''a''{{sub|4,3}} && ''a''{{sub|4,4}} && ''a''{{sub|4,5}} |( )}} |&&}}  yields the display style HTML+CSS parenthesized matrix A5,4 =  a1,1 a1,2 a1,3 a1,4 a1,5 a2,1 a2,2 a2,3 a2,4 a2,5 a3,1 a3,2 a3,3 a3,4 a3,5 a4,1 a4,2 a4,3 a4,4 a4,5 while the code {{indent}}{{math| \mathbf{A}_{5,4} {{=}} <!-- -->{{matrix| a_{1,1} && a_{1,2} && a_{1,3} && a_{1,4} && a_{1,5} \\ a_{2,1} && a_{2,2} && a_{2,3} && a_{2,4} && a_{2,5} \\ a_{3,1} && a_{3,2} && a_{3,3} && a_{3,4} && a_{3,5} \\ a_{4,1} && a_{4,2} && a_{4,3} && a_{4,4} && a_{4,5} |( )|tex}} |$$}}


yields the display style LaTeX parenthesized matrix

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

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}}

|$$}}  yields the display style LaTeX (rendered as .PNG image) parenthesized matrix ${\displaystyle {\begin{array}{l}\displaystyle {\mathbf {A} _{m,n}={\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,n}\\a_{2,1}&a_{2,2}&\cdots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\cdots &a_{m,n}\end{pmatrix}}}\end{array}}}$ ### Bracketed matrix The code {{indent}}{{math| '''A'''{{sub|5,4}} {{=|sp}} <!-- -->{{matrix| ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && ''a''{{sub|1,3}} && ''a''{{sub|1,4}} && ''a''{{sub|1,5}} \\ ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && ''a''{{sub|2,3}} && ''a''{{sub|2,4}} && ''a''{{sub|2,5}} \\ ''a''{{sub|3,1}} && ''a''{{sub|3,2}} && ''a''{{sub|3,3}} && ''a''{{sub|3,4}} && ''a''{{sub|3,5}} \\ ''a''{{sub|4,1}} && ''a''{{sub|4,2}} && ''a''{{sub|4,3}} && ''a''{{sub|4,4}} && ''a''{{sub|4,5}} |[ ]}} |&&}}  yields the display style HTML+CSS [square] bracketed matrix A5,4 =  a1,1 a1,2 a1,3 a1,4 a1,5 a2,1 a2,2 a2,3 a2,4 a2,5 a3,1 a3,2 a3,3 a3,4 a3,5 a4,1 a4,2 a4,3 a4,4 a4,5 while the code {{indent}}{{math| \mathbf{A}_{5,4} {{=}} <!-- -->{{matrix| a_{1,1} && a_{1,2} && a_{1,3} && a_{1,4} && a_{1,5} \\ a_{2,1} && a_{2,2} && a_{2,3} && a_{2,4} && a_{2,5} \\ a_{3,1} && a_{3,2} && a_{3,3} && a_{3,4} && a_{3,5} \\ a_{4,1} && a_{4,2} && a_{4,3} && a_{4,4} && a_{4,5} |[ ]|tex}} |$$}}


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

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

### 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}}

|$$}}  yields the display style LaTeX (rendered as .PNG image) matrix determinant ${\displaystyle {\begin{array}{l}\displaystyle {\operatorname {det} ~\mathbf {A} _{1,1}={\begin{vmatrix}a_{1,1}\end{vmatrix}}=|a_{1,1}|}\end{array}}}$ The code {{indent}}{{math| det '''A'''{{sub|''n'',''n''}} {{=|sp}} <!-- -->{{matrix| ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && {{...|cdots}} && ''a''{{sub|1,''n''}} \\ ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && {{...|cdots}} && ''a''{{sub|2,''n''}} \\ {{...|vdots}} && {{...|vdots}} && {{...|ddots}} && {{...|vdots}} \\ ''a''{{sub|''n'',1}} && ''a''{{sub|''n'',2}} && {{...|cdots}} && ''a''{{sub|''n'',''n''}} |det}} |&&}}  yields the display style HTML+CSS matrix determinant det An,n =  a1,1 a1,2 ⋯ a1,n a2,1 a2,2 ⋯ a2,n ⋮ ⋮ ⋱ ⋮ an,1 an,2 ⋯ an,n while the code {{indent}}{{math| \operatorname{det} ~ \mathbf{A}_{n,n} {{=}} <!-- -->{{matrix| a_{1,1} && a_{1,2} && \cdots && a_{1,n} \\ a_{2,1} && a_{2,2} && \cdots && a_{2,n} \\ \vdots && \vdots && \ddots && \vdots \\ a_{n,1} && a_{n,2} && \cdots && a_{n,n} |det|tex}} |$$}}


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

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

### 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}},

|$$}}{{nl}} where {{math|{{det|\mathbf{A}|tex}}|}} is the [[determinant]] of {{math|\mathbf{A}|}}, {{math|\mathbf{C}|}} is the [[matrix of cofactors]] of <!-- -->{{math|\mathbf{A}|}}, and {{math|\mathbf{C}{{sup|{\rm T}|tex}}|}} represents the matrix [[transpose]] of {{math|\mathbf{C}|}}.  yields the display style LaTeX matrix inverse formula ${\displaystyle {\begin{array}{l}\displaystyle {\mathbf {A} ^{\!-1}={\frac {\mathbf {C} ^{\rm {T}}{\!\,\!}}{|\mathbf {A} |}}={\frac {{\begin{pmatrix}C_{1,1}{\!\,\!}&C_{1,2}{\!\,\!}&\cdots &C_{1,n}{\!\,\!}\\C_{2,1}{\!\,\!}&C_{2,2}{\!\,\!}&\cdots &C_{2,n}{\!\,\!}\\\vdots &\vdots &\ddots &\vdots \\C_{n,1}{\!\,\!}&C_{n,2}{\!\,\!}&\cdots &C_{n,n}{\!\,\!}\end{pmatrix}}^{\rm {T}}{\!\,\!}}{\begin{vmatrix}a_{1,1}{\!\,\!}&a_{1,2}{\!\,\!}&\cdots &a_{1,n}{\!\,\!}\\a_{2,1}{\!\,\!}&a_{2,2}{\!\,\!}&\cdots &a_{2,n}{\!\,\!}\\\vdots &\vdots &\ddots &\vdots \\a_{n,1}{\!\,\!}&a_{n,2}{\!\,\!}&\cdots &a_{n,n}{\!\,\!}\end{vmatrix}}},}\end{array}}}$ where${\displaystyle \textstyle {|\mathbf {A} |}}$ is the determinant of${\displaystyle \textstyle {\mathbf {A} }}$,${\displaystyle \textstyle {\mathbf {C} }}$ is the matrix of cofactors of${\displaystyle \textstyle {\mathbf {A} }}$, and${\displaystyle \textstyle {\mathbf {C} ^{\rm {T}}{\!\,\!}}}$ represents the matrix transpose of${\displaystyle \textstyle {\mathbf {C} }}$. ### Nested matrices The code {{indent}}{{math| {{matrix| {{matrix| ''a'' && ''b'' \\ ''c'' && ''d'' }} && {{matrix| ''e'' && ''f'' \\ ''g'' && ''h'' }} \\ 0 && {{matrix| ''i'' && ''j'' \\ ''k'' && ''l'' }} }} |&&}}  yields the display style HTML+CSS  a b c d  e f g h 0  i j k l The code {{indent}}{{math| {{matrix| {{matrix| ''a'' && ''b'' \\ ''c'' && ''d'' |tex}} && {{matrix| ''e'' && ''f'' \\ ''g'' && ''h'' |tex}} \\ 0 && {{matrix| ''i'' && ''j'' \\ ''k'' && ''l'' |tex}} |tex}} |$$}}


yields the display style LaTeX

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