Determinants

A determinant is a scalar attached to a square matrix. Geometrically, it is the signed volume-scaling factor of the linear map. In two dimensions it scales oriented area; in three dimensions it scales oriented volume; in $n$ dimensions it scales $n$-dimensional volume.

The sign tracks orientation. A positive determinant preserves orientation, a negative determinant flips orientation, and a zero determinant means the map collapses space into a lower-dimensional set. That collapse is why determinants connect directly to rank, invertibility, and eigenvalues.

Defining math

For a $2\times 2$ matrix,

$$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\det (A)=ad-bc.$$

The columns of $A$ are where the basis vectors land. The determinant is the signed area of the parallelogram spanned by those two columns.

For a general $n\times n$ matrix, the determinant can be defined by expanding along any fixed row $i$:

$$\det (A)=\sum _{j=1}^n(-1{)}^{i+j}{a}_{ij}\det (A_{\hat{i},\hat{j}}).$$

Here $A_{\hat{i},\hat{j}}$ is the smaller matrix formed by deleting row $i$ and column $j$ from $A$. The factor $(-1{)}^{i+j}$ gives the checkerboard sign pattern

$$\left[\begin{array}{cccc}+& -& +& \cdots \\ -& +& -& \cdots \\ +& -& +& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right].$$

The term $\det (A_{\hat{i},\hat{j}})$ is called a minor, and $(-1{)}^{i+j}\det (A_{\hat{i},\hat{j}})$ is the corresponding cofactor. Intuitively, choosing $a_{ij}$ fixes how the selected row contributes; the minor measures the remaining volume scaling after that row and column have been removed. The alternating signs correct for orientation, just as the minus sign in the $2\times 2$ formula $ad-bc$ subtracts the oppositely oriented contribution.

The same definition can also be written in one compact permutation formula:

$$\det (A)=\sum _{\sigma \in S_n}\mathrm{sgn}(\sigma )\prod _{i=1}^n{A}_{i,\sigma (i)}.$$

Here $S_n$ is the set of all permutations of $\{1,\dots ,n\}$. A permutation $\sigma$ chooses exactly one column $\sigma (i)$ for each row $i$, so each product uses one entry from every row and every column. The sign $\mathrm{sgn}(\sigma )$ is $+1$ for an even permutation and $-1$ for an odd permutation. Equivalently, it records whether the column order chosen by $\sigma$ preserves or flips orientation.

These formulas are rarely the best way to compute large determinants directly, but they make the structure explicit: a determinant is a signed sum of full row-column matchings, and the signs enforce oriented volume.

Important consequences are:

$$\det (AB)=\det (A)\det (B),\det (A)=0⟺A\text{is singular}.$$

The second statement means that a square matrix is invertible exactly when its determinant is nonzero.

Worked example

Take

$$B=\left[\begin{array}{cc}2& 1\\ 1& 2\end{array}\right].$$

Its determinant is

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

So $B$ multiplies every oriented area by $3$. The first column $(2,1)$ is where $e_1$ lands, and the second column $(1,2)$ is where $e_2$ lands. The unit square becomes the parallelogram spanned by those two column vectors.

This is the same matrix used on Eigenvalues and Eigenvectors. There, the eigenvalues are $3$ and $1$; their product is also the determinant:

$$\det (B)=3\cdot 1=3.$$

For diagonalizable matrices, the determinant is the product of eigenvalues because each eigenvalue is a stretch factor along an eigenvector direction.

Intuition

The determinant answers: “How much does this square matrix scale full-dimensional volume?”

• $\det (A)=2$ doubles oriented volume.
• $\det (A)=-2$ doubles volume and flips orientation.
• $\det (A)=1$ preserves volume, though it may shear or rotate.
• $\det (A)=0$ collapses volume to zero, so some independent direction is lost.

That last case is the key connection to rank. If a matrix collapses a square into a line, or a cube into a plane, its columns are dependent and it cannot be inverted.

Caveats

Determinants are conceptually important but numerically blunt. In higher dimensions, determinants can become extremely large or small, and a nonzero determinant does not by itself say whether a matrix is well-conditioned. For numerical stability, singular values and condition numbers are usually more informative than determinant magnitude.

References

• MIT OpenCourseWare: 18.06 Linear Algebra

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