Determinant

Construction of Determinant, Part 1. For $n=2$

the determinant of $2 \times 2$ square matrix

(denoted by “ ${\det A}$ ”) is defined as

$\displaystyle \det A = \det \begin{bmatrix}a_{11} & a_{12} a_{21} & a_{22} \end{bmatrix} = a_{11} a_{22} - a_{12} a_{21}$

When $n\ge 3$ in general the determinant will be introduced recursively for an $n\times n$ square matrix. It starts with the submatrix $A_{ij}$ by “deleting the

th row and

th column” from an $n\times n$ square matrix

. Observe that $A_{ij}$ 's are $(n-1)\times (n-1)$ square matrices. Therefore, once the construction of ${\det A}$ is determined for $(n-1)\times (n-1)$ matrices, we can also introduce the (i,j)-cofactor $C_{ij}$ for $n\times n$

$\displaystyle C_{ij} = (-1)^{i+j} \det A_{ij}$ for $i,j = 1,\ldots,n$ . $\displaystyle$

Construction of Determinant, Part 2. For $n\ge 3$ the determinant of $n\times n$ matrix can be constructed recursively by

$\displaystyle \det A = a_{11} C_{11} + a_{12} C_{12} + a_{13} C_{13} + \cdots + a_{1n} C_{1n} .$

where the cofactors are calculated by applying the determinant for $(n-1)\times (n-1)$ square matrices $A_{ij}$ 's.

In Matlab/Octave the function det(A) is called to compute the determinant for a square matrix .

EXAMPLE 1. Compute the determinant of $A = \begin{bmatrix} 1 & 5 & -2 \\ 2 & 4 &-1 \\ 0 &-2 & 0 \end{bmatrix}$