Determinant

For a $ 2 \times 2$ matrix $ A$, the determinant of $ A$ (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}
$

Let $ A$ be an $ n\times n$ square matrix in general. Then we can define the submatrix by ``deleting the $ i$th row and $ j$th column of $ A$'', denoted by $ A_{ij}$. Observing that $ A_{ij}$'s are $ (n-1)\times (n-1)$ square matrices, the determinant of $ A$ is given recursively by

$\displaystyle \det A
= a_{11}\det A_{11}
- a_{12}\det A_{12}
+ a_{13}\det A_{13}
- \cdots + (-1)^{1+n} a_{1n}\det A_{1n}.
$

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

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



Subsections


Department of Mathematics
Last modified: 2005-10-21