Determinants and Row Operations

Let $M\in M_{n\times n}(\mathbb{R})$ Suppose $M^{\prime }$ is obtained from $M$ by row operation

1. Scale a row. if $M\stackrel{\lambda R_i\to R_i}{\to }{M}^{\prime }$ then $\det (M^{\prime })=\lambda \det (M)$
2. Exchange a row $R_i,R_j$ $M\stackrel{R_i\leftrightarrow R_j}{\to }{M}^{\prime }$ then $\det (M^{\prime })=-\det (M)$
3. Add a row $R_i,R_j$ $M\stackrel{\lambda R_i+R_j\to R_j}{\to }{M}^{\prime }$ then $\det (M^{\prime })=\det (M)$

Proofs

Proving Scalability

$\det (M^{\prime })=\det (R_1,\dots ,\lambda R_i,\dots ,R_n)$ by multilinearity $=\lambda \det (R_1,\dots ,R_i,\dots ,R_n)$

Proving Exchangability

$\det (M^{\prime })=\det (R_1,\dots ,R_j,\dots ,R_i,\dots ,R_n)$ $=-\det (R_1,\dots ,R_i,\dots ,R_j,\dots R_n)$ by Alternating

Proving Additivity

$\det (M^{\prime })=\det (R_1,\dots ,R_j+\lambda R_i,\dots ,R_n)$ $=\det (R_1,\dots ,R_j,\dots ,R_n)+\lambda \det (R_1,..,R_i,\dots ,R_n)$ $=\det (M)$