Row Properties of the Determinant

Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$, and let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure. Let $A$ be a real $n\times n$ matrix, with entry notation as there. Let $S_{n}$ be the set of permutations of $[n]$, with the identity $\mathrm{id}$ and the composition of Permutations of an Initial Segment Form a Group under Composition, and let $\mathrm{sgn}$ be the sign, with the properties recorded in The Sign of a Permutation is Multiplicative.

The determinant $\det A$ is that of Determinant of a Real Square Matrix. In the formula defining it, the sum over $\sigma\in S_{n}$ is the sum over a finite index set of Sum over a Finite Index Set, the set $S_{n}$ being nonempty and finite by claim 4 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets, and the product of the entries $A_{1\sigma(1)},\dots,A_{n\sigma(n)}$ is the finite product over $[n]$, so that

$$\det A=\sum_{\sigma\in S_{n}}\mathrm{sgn}(\sigma)\prod_{i=1}^{n}A_{i\,\sigma(i)}.$$

Sums with a numerical index range are the finite sums of $\mathbb{R}$, powers are those of Natural Number Power of an Element of a Field, the identity matrix $I_{n}$ is that of Identity Matrix, the scalar multiple $\mu A$ is that of Scalar Multiple of a Real Matrix, and the transpose $A^{\top}$ is that of Transpose of a Real Matrix.

Then the following hold.

1. (Identity matrix) $\det I_{n}=1$.

2. (Linearity in one row) Let $i\in[n]$, let $m$ be a natural number, let $U$ be a real $m\times n$ matrix, and let $c$ be an $m$-tuple of real numbers, with components $c_{l}$, such that

$$A_{ij}=\sum_{l=1}^{m}c_{l}\,U_{lj}\qquad\text{for every }j\in[n].$$

For $l\in[m]$ let $A^{(l)}$ be the real $n\times n$ matrix with $A^{(l)}_{ij}=U_{lj}$ for every $j\in[n]$ and $A^{(l)}_{kj}=A_{kj}$ for every $k\in[n]$ with $k\ne i$ and every $j\in[n]$. Then

$$\det A=\sum_{l=1}^{m}c_{l}\,\det A^{(l)}.$$

3. (Permuting the rows) Let $\pi\in S_{n}$ and let $A_{\pi}$ be the real $n\times n$ matrix with $(A_{\pi})_{ij}=A_{\pi(i)\,j}$ for all $i,j\in[n]$. Then

$$\det A_{\pi}=\mathrm{sgn}(\pi)\,\det A.$$

4. (Two equal rows) If $p,q\in[n]$ satisfy $p\ne q$ and $A_{pj}=A_{qj}$ for every $j\in[n]$, then $\det A=0$.

5. (A zero row) If $i\in[n]$ satisfies $A_{ij}=0$ for every $j\in[n]$, then $\det A=0$.

6. (Dependent rows) If $x$ is an $n$-tuple of real numbers, with components $x_{i}$, such that $x_{k}\ne0$ for at least one $k\in[n]$ and

$$\sum_{i=1}^{n}x_{i}\,A_{ij}=0\qquad\text{for every }j\in[n],$$

then $\det A=0$.

7. (Scalar multiples) $\det(\mu A)=\mu^{n}\det A$ for every $\mu\in\mathbb{R}$.

8. (Transpose) $\det\bigl(A^{\top}\bigr)=\det A$.