Determinants

Precise definition

The determinant assigns a scalar $\det A$ to a square matrix. It is multilinear in the rows or columns, changes sign when two are swapped, and equals 1 on the identity. Geometrically, $|\det A|$ is the factor by which the associated linear map scales $n$-dimensional volume; the sign records orientation.

Notation and mathematical language

For $2\times2$, $\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc$. Row replacement leaves the determinant unchanged, swapping rows negates it, and scaling one row by $k$ scales the determinant by $k$. Also $\det(AB)=\det A\det B$ and $\det A^T=\det A$.

Conceptual picture

A zero determinant means the columns collapse dimension and are linearly dependent. Equivalent statements for a square matrix include invertibility, full rank, a pivot in every row and column, and only the trivial solution to $A\mathbf x=0$.

Conditions and key results

Determinants are defined for square matrices. Cofactor expansion includes signs $(-1)^{i+j}$ and is best along a sparse row or column. Row operations can compute determinants efficiently only when their effect is tracked.

A reliable strategy

1. Check the matrix is square and choose a method suited to its size and structure.
2. For small matrices use a formula or cofactor expansion; for larger ones reduce to triangular form while recording row effects.
3. Multiply diagonal entries after triangularization and undo accumulated scaling or swap factors.
4. Interpret zero/nonzero status, sign, and magnitude in the context and verify with another property.

Fully worked example

Interpretation and application

Determinants appear in change of variables, orientation, eigenvalue products, and uniqueness tests. A large determinant does not necessarily mean a matrix is numerically well-conditioned because rescaling units changes it.

Common mistakes

Verification and reasonableness

• Compute via a second expansion or tracked row reduction.
• Check $\det A\ne0$ against full pivot count.
• For a product, compare direct computation with $\det A\det B$.

Practice

1. Find $\det\begin{pmatrix}2&3\\1&4\end{pmatrix}$.
2. What does one row swap do?
3. What does determinant zero say about columns?

Further deduction

The determinant is alternating: if two columns are equal, swapping them changes the sign but leaves the matrix unchanged, so $\det A=-\det A$ and hence $\det A=0$. Multilinearity extends this to any dependent columns. This short structural proof explains the invertibility test more deeply than memorizing a list of equivalent statements.

Laplace expansion along any row or column gives the same determinant because it is a theorem derived from the alternating multilinear definition. For triangular $A$, only the identity permutation contributes, so $\det A$ is the product of diagonal entries. Row reduction seeks this structure, explaining rather than merely stating the diagonal-product rule.

Related topics

• Inverse Matrix
• Eigenvalues
• Cramer's Rule