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Determinant

The determinant is a function that assigns to each square matrix $ A \in M_{n\times n}(\mathbb{R}) $ (or over any field) a real number:

$$\det : M_{n\times n} \to \mathbb{R}$$

It is defined only for square matrices.

Geometric interpretation

The determinant measures the volume scaling factor of the linear transformation associated with the matrix.

If $ T(\mathbf{x}) = A\mathbf{x} $, then:

In $ \mathbb{R}^2 $, $ |\det(A)| $ is the factor by which area is scaled.
In $ \mathbb{R}^3 $, $ |\det(A)| $ is the factor by which volume is scaled.
In dimension $ n $, it scales $n$-dimensional volume.

Moreover:

If $ \det(A) > 0 $, orientation is preserved.
If $ \det(A) < 0 $, orientation is reversed.
If $ \det(A) = 0 $, space is collapsed (dimension is lost).

This yields a fundamental criterion:

$\det(A)=0 \quad \Longleftrightarrow \quad A $ is not invertible

Algebraic interpretation

The determinant helps decide:

Whether a linear system has a unique solution.
Whether a set of vectors is linearly independent.
Whether a matrix is invertible.
Whether a change of basis is valid.

For a system $ A\mathbf{x}=\mathbf{b} $:

$\det(A)\neq 0 \Rightarrow $unique solution

Fundamental properties

Important structural properties:

$\det(I)=1$
$\det(AB)=\det(A)\det(B)$
$\det(A^{-1})=\frac{1}{\det(A)}$ if $A$ is invertible
Swapping two rows changes the sign of the determinant.
If one row is a linear combination of the others, the determinant is zero.
If two rows are equal, the determinant is zero.
If $A$ is triangular, then:

$\det(A)=$ product of the diagonal entries

Classical formulas

The $2\times 2$ case

$$A = \begin{pmatrix} a & b\\ c & d \end{pmatrix} \Rightarrow\det(A)=ad-bc$$

Interpretation: the (oriented) area of the parallelogram generated by the column vectors.

The general case

A classical definition uses cofactor expansion:

$$\det(A)=\sum_{j=1}^{n} (-1)^{i+j} a_{ij}\,\det(M_{ij})$$

An equivalent definition uses permutations:

$$\det(A)=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\prod_{i=1}^{n} a_{i,\sigma(i)}$$

This latter formula connects naturally with group theory (through $S_n$).

Instructions

In this applet, determinant calculation exercises are generated randomly. Using the green slider, you can choose the order of the square matrix, which may be 2, 3, or 4. Compute the determinant of the displayed matrix and enter your answer in the corresponding field. Then click the “Check” button. If your answer is correct, the app will confirm it. If it is incorrect, you will receive feedback indicating whether the correct determinant is greater than or less than the value you entered. If you make a mistake, you can recalculate the determinant and check your new answer. To generate a new matrix with different values, click the “New Exercise” button.

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