Manuals/calci/DET

DET(array)

• $array$ is the set of numbers.

Description

• This function gives the determinant value of a matrix.
• To calculate the determinant of a matrix, we can choose only square matrix.i.e. Number of rows and number of columns should be equal.
• Determinant of the identity matrix is always 1.
• Determinant of the matrix $A$ 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 $j^{th}$ column and $i=1$ to $n$ & $j=1$ to $n$.
• For any $i$ and $j$, set $A_{ij}$ (called the co-factors), then the general formula for determinant of the matrix $A$ is,

$|A|=\sum _{j=1}^{n}a_{ij}A_{ij}$, for any fixed $i$. Also$|A|=\sum _{i=1}^{n}a_{ij}A_{ij}$, for any fixed $j$.

• This function will give the result as error when

1. Any one of the element in array is empty or contain non-numeric
2. Number of rows is not equal to number of columns

Examples

1. =DET([[6,4,8],[3,6,1],[2,4,5]]) = 104
2. =DET([[-5,10],[6,-8]]) = -20
3. =DET([[1,0,2,1],[4,0,2,-1],[1,4,5,2],[3,1,2,0]]) = 17
4. =DET([1,2,3],[5,2,8]) = NAN

See Also

• MINVERSE
• MMULT

References

• Determinant

• List of Main Z Functions
• Z3 home