Manuals/calci/MATRIXDETERMINANT

MATRIXDETERMINANT (a)

• $a$ is any square matrix.

Description

• This function is calculating the determinant of the given matrix.
• In $MATRIXDETERMINANT(a)$, $a$ is any square matrix.
• The determinant of a matrix is a special number that can be calculated from a square matrix.
• The determinant of a matrix is denoted by $det(A)$ or $|A|$.
• Let $A$ be 2x2 matrix with the elements

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

• Then $det(A)=ad-bc$, where $a,b,c,d$ all are real numbers.
• Let $A$ be the 3x3 matrix with the elements

$A={\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\\\end{bmatrix}}$ Then $|A|=a{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c{\begin{vmatrix}d&e\\g&h\end{vmatrix}}$: $|A|=a(ei-fh)-b(di-fg)+c(dh-eg)$

• Let $A$ be a square matrix of order $n$. Write $A=(a_{ij})$,
• Where $a_{ij}$ is the entry on the $i^{th}$ row and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{th}} column and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} & Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j=1} to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} .
• For any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{ij}} (called the co-factors), then the general formula for determinant of the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} is,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|=\sum_{j=1}^n a_{ij} A_{ij}} , for any fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} . AlsoFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A|=\sum_{i=1}^n a_{ij} A_{ij}} , for any fixed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} .

Examples

1. MATRIXDETERMINANT([[6,4,8],[3,6,1],[2,4,5]]) = 104.00000000000001
2. MATRIXDETERMINANT([[8,-4],[12,5]]) =88
3. MATRIXDETERMINANT([1,2,3,4;5,6,-7,8;12,10,-13,15;11,7,5,3]) = 1514

See Also

• MINVERSE
• MMULT

References

• Determinant

• List of Main Z Functions
• Z3 home