Laplace expansions

Define the $ (i,j)$-cofactor $ C_{ij}$ of $ A$ by

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

Then the determinant of $ A$ can be expanded in any of the following expansions, often referred as Laplace expansions:
$\displaystyle \det A$ $\displaystyle =$ $\displaystyle a_{i1} C_{i1} + a_{i2} C_{i2} + \cdots + a_{in} C_{in}$   $\displaystyle \mbox{ for $i = 1,\ldots,n$; }$ (3)
  $\displaystyle =$ $\displaystyle a_{1j} C_{1j} + a_{2j} C_{2j} + \cdots + a_{nj} C_{nj}$   $\displaystyle \mbox{ for $j = 1,\ldots,n$. }$ (4)

The Laplace expansion with $ i = j = 1$ immediately indicate that the transpose has no effect in determinant, that is, that

$\displaystyle \det A^T = \det A.$ (5)

Example 2 Use the Laplace expansion across the third row to compute $ det A$ for $ A = \begin{bmatrix}
1 & 5 & 0 \\
2 & 4 &-1 \\
0 &-2 & 0
\end{bmatrix}$



Department of Mathematics
Last modified: 2005-10-21