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Template:Matrix/doc
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.
Contents
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 =
|
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}} |$$}}
yields the display style LaTeX raw matrix
Parenthesized matrix
The code
{{indent}}{{math| '''A'''{{sub|5,4}} {{=|sp}} <!-- -->{{matrix| ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && ''a''{{sub|1,3}} && ''a''{{sub|1,4}} && ''a''{{sub|1,5}} \\ ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && ''a''{{sub|2,3}} && ''a''{{sub|2,4}} && ''a''{{sub|2,5}} \\ ''a''{{sub|3,1}} && ''a''{{sub|3,2}} && ''a''{{sub|3,3}} && ''a''{{sub|3,4}} && ''a''{{sub|3,5}} \\ ''a''{{sub|4,1}} && ''a''{{sub|4,2}} && ''a''{{sub|4,3}} && ''a''{{sub|4,4}} && ''a''{{sub|4,5}} |( )}} |&&}}
yields the display style HTML+CSS parenthesized matrix
A5,4 =
|
while the code
{{indent}}{{math| \mathbf{A}_{5,4} {{=}} <!-- -->{{matrix| a_{1,1} && a_{1,2} && a_{1,3} && a_{1,4} && a_{1,5} \\ a_{2,1} && a_{2,2} && a_{2,3} && a_{2,4} && a_{2,5} \\ a_{3,1} && a_{3,2} && a_{3,3} && a_{3,4} && a_{3,5} \\ a_{4,1} && a_{4,2} && a_{4,3} && a_{4,4} && a_{4,5} |( )|tex}} |$$}}
yields the display style LaTeX parenthesized matrix
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 =
|
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}} |$$}}
yields the display style LaTeX (rendered as .PNG image) parenthesized matrix
Bracketed matrix
The code
{{indent}}{{math| '''A'''{{sub|5,4}} {{=|sp}} <!-- -->{{matrix| ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && ''a''{{sub|1,3}} && ''a''{{sub|1,4}} && ''a''{{sub|1,5}} \\ ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && ''a''{{sub|2,3}} && ''a''{{sub|2,4}} && ''a''{{sub|2,5}} \\ ''a''{{sub|3,1}} && ''a''{{sub|3,2}} && ''a''{{sub|3,3}} && ''a''{{sub|3,4}} && ''a''{{sub|3,5}} \\ ''a''{{sub|4,1}} && ''a''{{sub|4,2}} && ''a''{{sub|4,3}} && ''a''{{sub|4,4}} && ''a''{{sub|4,5}} |[ ]}} |&&}}
yields the display style HTML+CSS [square] bracketed matrix
A5,4 =
|
while the code
{{indent}}{{math| \mathbf{A}_{5,4} {{=}} <!-- -->{{matrix| a_{1,1} && a_{1,2} && a_{1,3} && a_{1,4} && a_{1,5} \\ a_{2,1} && a_{2,2} && a_{2,3} && a_{2,4} && a_{2,5} \\ a_{3,1} && a_{3,2} && a_{3,3} && a_{3,4} && a_{3,5} \\ a_{4,1} && a_{4,2} && a_{4,3} && a_{4,4} && a_{4,5} |[ ]|tex}} |$$}}
yields the display style LaTeX (rendered as .PNG image) [square] bracketed matrix
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 =
|
while the code
{{indent}}{{math| \operatorname{det} ~ \mathbf{A}_{1,1} {{=}} <!-- -->{{matrix| a_{1,1} |det|tex}} <!-- -->{{=}} {{det| a_{1,1} |tex}} |$$}}
yields the display style LaTeX (rendered as .PNG image) matrix determinant
The code
{{indent}}{{math| det '''A'''{{sub|''n'',''n''}} {{=|sp}} <!-- -->{{matrix| ''a''{{sub|1,1}} && ''a''{{sub|1,2}} && {{...|cdots}} && ''a''{{sub|1,''n''}} \\ ''a''{{sub|2,1}} && ''a''{{sub|2,2}} && {{...|cdots}} && ''a''{{sub|2,''n''}} \\ {{...|vdots}} && {{...|vdots}} && {{...|ddots}} && {{...|vdots}} \\ ''a''{{sub|''n'',1}} && ''a''{{sub|''n'',2}} && {{...|cdots}} && ''a''{{sub|''n'',''n''}} |det}} |&&}}
yields the display style HTML+CSS matrix determinant
det An,n =
|
while the code
{{indent}}{{math| \operatorname{det} ~ \mathbf{A}_{n,n} {{=}} <!-- -->{{matrix| a_{1,1} && a_{1,2} && \cdots && a_{1,n} \\ a_{2,1} && a_{2,2} && \cdots && a_{2,n} \\ \vdots && \vdots && \ddots && \vdots \\ a_{n,1} && a_{n,2} && \cdots && a_{n,n} |det|tex}} |$$}}
yields the display style LaTeX (rendered as .PNG image) matrix determinant
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 |
A − 1 =
|
where
| A | |
A |
C |
A |
CT |
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}}, |$$}}{{nl}} where {{math|{{det|\mathbf{A}|tex}}|$}} is the [[determinant]] of {{math|\mathbf{A}|$}}, {{math|\mathbf{C}|$}} is the [[matrix of cofactors]] of <!-- -->{{math|\mathbf{A}|$}}, and {{math|\mathbf{C}{{sup|{\rm T}|tex}}|$}} represents the matrix [[transpose]] of {{math|\mathbf{C}|$}}.
yields the display style LaTeX matrix inverse formula
where is the determinant of, is the matrix of cofactors of, and represents the matrix transpose of.
Nested matrices
The code
{{indent}}{{math| {{matrix| {{matrix| ''a'' && ''b'' \\ ''c'' && ''d'' }} && {{matrix| ''e'' && ''f'' \\ ''g'' && ''h'' }} \\ 0 && {{matrix| ''i'' && ''j'' \\ ''k'' && ''l'' }} }} |&&}}
yields the display style HTML+CSS
|
The code
{{indent}}{{math| {{matrix| {{matrix| ''a'' && ''b'' \\ ''c'' && ''d'' |tex}} && {{matrix| ''e'' && ''f'' \\ ''g'' && ''h'' |tex}} \\ 0 && {{matrix| ''i'' && ''j'' \\ ''k'' && ''l'' |tex}} |tex}} |$$}}
yields the display style LaTeX
See also
- {{det}} (matrix determinant template)