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 $j^{th}$ 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 $j$.