10 Properties of Determinants

Determinant properties exist to enable faster, more robust computation—and also help determine whether a matrix is singular (degenerate) or whether a linear transformation preserves information.

I’ll annotate each elementary row operation with its effect on the determinant, ensuring no sign changes or scalar multipliers are overlooked.

In the previous article, we introduced the formal definition of determinants. This article continues by organizing several key determinant properties—tools that prove indispensable both for conceptual understanding and practical computation, especially in AI and machine learning applications, where linear algebra fundamentals are essential.

1. Fundamental Properties of Determinants

Below are the most important determinant properties:

When learning determinant properties, first connect them to row/column operations, invertibility, and matrix multiplication. These properties are used far more frequently than direct cofactor expansion.

1.1 Swapping Two Rows (or Columns) Flips the Sign

Swapping any two rows (or two columns) of a matrix $A$ flips the sign of its determinant. Mathematically:

If $B$ is obtained from $A$ by swapping two rows (or two columns), then:

$$\det(B) = -\det(A)$$

Example:

Consider the matrix

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

Its determinant is:

$\det(A) = 1 \cdot 4 - 2 \cdot 3 = 4 - 6 = -2$

Now swap rows 1 and 2 to obtain

$B = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}$

Then:

$\det(B) = 3 \cdot 2 - 4 \cdot 1 = 6 - 4 = 2$

Indeed, $\det(B) = -\det(A)$.

1.2 Determinant of a Matrix with a Zero Row (or Column) Is Zero

If any row (or column) of matrix $A$ consists entirely of zeros, then $\det(A) = 0$.

If $A$ has an all-zero row or column, then:

$$\det(A) = 0$$

Example:

Let

$C = \begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix}$

Then:

$\det(C) = 1 \cdot 0 - 2 \cdot 0 = 0$

As expected, the determinant is zero.

1.3 Scalar Multiplication of a Row (or Column)

Multiplying a single row (or column) of matrix $A$ by a scalar $k$ multiplies the determinant by $k$.

If $B$ is obtained from $A$ by multiplying one row (or column) by $k$, then:

$$\det(B) = k \cdot \det(A)$$

Example:

Let

$D = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$

Multiply row 1 by 2 to get

$E = \begin{pmatrix} 2 & 4 \\ 3 & 4 \end{pmatrix}$

Compute:

$\det(D) = 1 \cdot 4 - 2 \cdot 3 = -2$
$\det(E) = 2 \cdot 4 - 4 \cdot 3 = 8 - 12 = -4$

Indeed, $\det(E) = 2 \cdot \det(D)$.

1.4 Adding a Multiple of One Row (or Column) to Another Leaves the Determinant Unchanged

Adding a scalar multiple of one row (or column) to another row (or column) does not change the determinant.

If $B$ is obtained from $A$ by adding $k$ times row $i$ to row $j$ ($i \neq j$), then:

$$\det(B) = \det(A)$$

Example:

Take matrix $A$ again and add twice row 2 to row 1:

$F = \begin{pmatrix} 1 + 2 \cdot 3 & 2 + 2 \cdot 4 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 7 & 10 \\ 3 & 4 \end{pmatrix}$

Then:

$\det(F) = 7 \cdot 4 - 10 \cdot 3 = 28 - 30 = -2$

Thus, $\det(F) = \det(A)$.

2. Dimensional Properties of Determinants

2.1 Matrix Dimensions and Determinants

The topic “Properties of Determinants” can be read through four lenses: context, concept, action, and outcome. Align these four dimensions first—then revisit parameters, code, or workflows in the main text.

Determinants are defined only for square matrices; therefore, determinants of non-square matrices are undefined. Consequently, all determinant properties discussed herein apply exclusively to square matrices.

2.2 Recursive Definition of Determinants

For an $n \times n$ matrix $A$, the determinant can be computed recursively via cofactor expansion along any row or column—reducing the problem to computing $n$ determinants of $(n-1) \times (n-1)$ submatrices. For example, expanding along the first row yields:

$\det(A) = \sum_{j=1}^n (-1)^{1+j} a_{1j} \det(A_{1j})$

where $A_{1j}$ denotes the $(n-1)\times(n-1)$ submatrix formed by deleting row 1 and column $j$ from $A$.

After studying “Properties of Determinants,” try applying them in your own context. Focus especially on whether inputs, transformations, and outputs align coherently.

To apply “Properties of Determinants” to your own task, begin by narrowing the scope—test just one critical decision point.

3. Summary

Determinant properties provide powerful tools for computing and interpreting matrices in linear algebra. In the next article, we’ll explore concrete methods for evaluating determinants—laying essential groundwork for more advanced topics such as solving linear systems and computing eigenvalues, both of which underpin numerous AI algorithms. We hope that mastering these properties deepens your intuition and broadens your ability to apply determinants meaningfully!