Matrix Algebra

Column Operations

Properties of 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.

$\det [ \mathbf{a}_1 \ldots (\mathbf{a}_j + c \mathbf{a}_i) \ldots \ma... ...a}_n] = \det [ \mathbf{a}_1 \ldots \mathbf{a}_j \ldots \mathbf{a}_n]$ (for $i \neq j$ ).
$\det [ \mathbf{a}_1 \ldots \mathbf{a}_k \ldots \mathbf{a}_{k'} \ldot... ...{a}_1 \ldots \mathbf{a}_{k'} \ldots \mathbf{a}_k \ldots \mathbf{a}_n]$
$\det [ \mathbf{a}_1 \ldots c \mathbf{a}_j \ldots \mathbf{a}_n] = c\cdot\det [ \mathbf{a}_1 \ldots \mathbf{a}_j \ldots \mathbf{a}_n]$
$\det [ \mathbf{a}_1 \ldots (\mathbf{a}_j + \mathbf{b}_j) \ldots \mathbf{a}_n]$ $= \det [ \mathbf{a}_1 \ldots \mathbf{a}_j \ldots \mathbf{a}_n] + \det [ \mathbf{a}_1 \ldots \mathbf{b}_j \ldots \mathbf{a}_n]$ (where $\mathbf{b}_j$ is another column vector operated on the th column of ).
$\det [ \mathbf{a}_1 \ldots \mathbf{a}_k \ldots \mathbf{a}_{k} \ldots \mathbf{a}_n] = 0$

Further discussion of the properties. Properties 3 and 4 can be immediately verified by the Laplace expansions. Property 1 is implied by Properties 4 and 5. Finally we outline how to prove Properties 2 and 5 together by induction: When we apply these forms of expansion 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

, we have $\det B = 0$ , which shows Property 5.