Column operations

Let $ A = [ \mathbf{a}_1  \mathbf{a}_2  \ldots  \mathbf{a}_n]$ be an $ n\times n$ square matrix composed of column vectors  $ \mathbf{a}_1,
\mathbf{a}_2, \ldots ,\mathbf{a}_n$. Then we have the following properties for determinants.

  1. $ \det [ \mathbf{a}_1 
\ldots (\mathbf{a}_j + c \mathbf{a}_i) 
\ldots \mathbf{a}_n]
=
\det [ \mathbf{a}_1 
\ldots \mathbf{a}_j 
\ldots \mathbf{a}_n]$ (for $ i \neq j$).

  2. $ \det [ \mathbf{a}_1 
\ldots \mathbf{a}_k \ldots \mathbf{a}_{k'} 
\ldots\...
...bf{a}_1 
\ldots \mathbf{a}_{k'} \ldots \mathbf{a}_k 
\ldots \mathbf{a}_n]$

  3. $ \det [ \mathbf{a}_1 
\ldots c \mathbf{a}_j 
\ldots \mathbf{a}_n]
=
c\cdot\det [ \mathbf{a}_1 
\ldots \mathbf{a}_j 
\ldots \mathbf{a}_n]$

  4. $ \det [ \mathbf{a}_1 
\ldots (\mathbf{a}_j + \mathbf{b}_j) 
\ldots \mathbf...
...hbf{a}_n]
+
\det [ \mathbf{a}_1 
\ldots \mathbf{b}_j 
\ldots \mathbf{a}_n]$

    (where $ \mathbf{b}_j$ is another column vector operated on the $ j$th column of $ A$).

  5. $ \det [ \mathbf{a}_1 
\ldots \mathbf{a}_k \ldots \mathbf{a}_{k} 
\ldots \mathbf{a}_n]
= 0$

Properties 3 and 4 can be immediately verified by Laplace expansions. When we apply Laplace expansions recursively for $ j \neq k, k'$, we end up with the determinants of $ 2 \times 2$ matrices of the respective forms

$\displaystyle B = \begin{bmatrix}a_{ik} & a_{ik'}  a_{i'k} & a_{i'k'} \end{bmatrix}$    and $\displaystyle \quad
B' = \begin{bmatrix}a_{ik'} & a_{ik}  a_{i'k'} & a_{i'k} \end{bmatrix}.
$

If $ k \neq k'$, we have $ \det B = -\det B'$, which shows Property 2. If $ k = k$, we have $ \det B = 0$, which shows Property 5. Finally Properties 4 and 5 together implies Property 1.



Department of Mathematics
Last modified: 2005-10-21