Determinant

Every square matrix has a single number attached to it called its determinant, written $\det(A)$ or $|A|$. The picture to keep in mind is geometric: a square matrix acts on space as a linear transformation $\mathbf{x} \mapsto A\mathbf{x}$ — stretching, rotating, shearing, or flipping it — and the determinant is the factor by which areas (or volumes) get scaled under that transformation.

A few things follow immediately from that picture:

• If $\det(A) = 3$, every shape comes out three times bigger after applying $A$.
• The sign records orientation: positive means $A$ preserves it, negative means $A$ flips it (like a mirror reflection).
• $\det(A) = 0$ is the degenerate case — $A$ squashes space down into a lower-dimensional shadow with no volume left, which is exactly when $A$ has no inverse. So "$\det(A) \neq 0$" is the standard test for invertibility.

For the small cases that come up most often — $2 \times 2$ and $3 \times 3$ — that scale factor has a clean closed-form formula.

2×2 Determinant

A $2 \times 2$ matrix has two columns, each a vector in the plane. Together they span a parallelogram, and the determinant is just the signed area of that parallelogram.

The recipe to remember: multiply along the main diagonal ($a \cdot d$, top-left to bottom-right), then subtract the product along the anti-diagonal ($b \cdot c$, top-right to bottom-left).

3×3 Determinant

The story is the same, one dimension up. A $3 \times 3$ matrix has three columns — three vectors in space — and they span a parallelepiped (a slanted box). The determinant is the signed volume of that box.

The “cross out the row and column” leftover block has a name: its determinant is the minor of the entry, and the signed minor (with the checkerboard sign attached) is the cofactor. The alternating $+, -, +$ pattern is the same one that shows up in the cross-product determinant mnemonic.

Nothing forces you to walk the first row — expanding along any row or column gives the same answer, as long as you use the right sign pattern: the checkerboard $(-1)^{i+j}$ at position $(i, j)$. Picking a row or column with zeros saves effort, since those terms drop out before you ever compute the corresponding minor.

Determinant from Eigenvalues

Beyond the closed-form formulas above, the determinant has a second face that holds for every square matrix: it equals the product of the matrix’s eigenvalues. The $2 \times 2$ case shows why concretely, and the same argument lifts to any size.

Expanding $\det(A - \lambda I)$ for a $2 \times 2$ matrix gives the characteristic polynomial

Its roots are the eigenvalues $\lambda_1, \lambda_2$, so the same polynomial must also factor as:

Matching coefficients (Vieta’s formulas) reads off two classical identities at once:

For an $n \times n$ matrix the picture is the same: $\det(A)$ is the product of all $n$ eigenvalues, and $\operatorname{tr}(A)$ is their sum.

Key Properties

For square matrices $A, B \in \mathbb{R}^{n \times n}$ and a scalar $\lambda \in \mathbb{R}$:

• Multiplicativity: $\det(AB) = \det(A) \det(B)$ — composing two transformations multiplies their volume scale factors, exactly as you’d expect.
• Transpose invariance: $\det(A^\top) = \det(A)$ — so cofactor expansion along any row or column gives the same result.
• Scaling: $\det(\lambda A) = \lambda^n \det(A)$ for an $n \times n$ matrix — every column gets stretched by $\lambda$, and there are $n$ of them, so the volume picks up a factor of $\lambda^n$.
• Invertibility: $A$ is invertible if and only if $\det(A) \neq 0$ — a transformation that crushes volume to zero cannot be undone.
• Diagonal and triangular matrices: the determinant of a diagonal matrix — and more generally any upper- or lower-triangular matrix — is the product of its diagonal entries: $\det(D) = d_1 d_2 \cdots d_n$. In particular, $\det(I_n) = 1$, which makes sense: the identity transformation leaves every shape exactly as it found it.