Properties of Determinant

The determinant was defined via expansion along the first row. That definition is logically complete but computationally expensive — expanding an $n \times n$ matrix naively requires $n!$ multiplications. This article develops the properties that make determinants tractable: expansion along any row or column, the way row operations interact with det, and the identity $\det(AB) = \det(A)\det(B)$. It ends with two classical applications: the adjugate-based inverse formula and Cramer’s rule.

Laplace expansion along any row or column

The definition in Determinant expanded along row 1. The same value emerges from any row or column.

Theorem (Laplace expansion): For any $i \in \{1, \ldots, n\}$,

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

and for any $j \in \{1, \ldots, n\}$,

\det(A) = \sum_{i=1}^{n} (-1)^{i+j}\, a_{ij}\, \det(A_{ij}). \tag{2}

Here $A_{ij}$ is the $(n-1) \times (n-1)$ minor obtained by deleting row $i$ and column $j$, and $C_{ij} \coloneqq (-1)^{i+j} \det(A_{ij})$ is the $(i,j)$ cofactor.

In practice: expand along the row or column with the most zeros — each zero term contributes nothing and shrinks the work significantly.

Worked example

$A = \begin{pmatrix} 0 & 1 & 0 \\ 2 & 3 & 4 \\ 5 & 6 & 7 \end{pmatrix}.$

Expanding along row 1, only the $a_{12} = 1$ term is nonzero:

$\det(A) = (-1)^{1+2} \cdot 1 \cdot \det\!\begin{pmatrix}2 & 4 \\ 5 & 7\end{pmatrix} = -1 \cdot (14 - 20) = 6.$

Transpose

\det(A^\top) = \det(A). \tag{3}

Because the determinant is unchanged by transposition, any statement about rows holds equally for columns. In particular, column-expansion (equation (2)) follows from row-expansion (equation (1)) applied to $A^\top$, and every row-operation rule below has an exact column analogue.

The effect of row operations on det

Row operations — the building blocks of Gauss-Jordan elimination — interact with the determinant in a clean way:

Row operation	Effect on $\det(A)$
Swap two rows	Multiplies $\det$ by $-1$
Multiply row $i$ by scalar $c \ne 0$	Multiplies $\det$ by $c$
Add a scalar multiple of one row to another	$\det$ unchanged

These three rules turn Gaussian elimination into an efficient determinant algorithm. Apply Gaussian elimination to $A$ (using only row swaps and row additions — no row scaling), tracking the number of swaps $s$. The result is an upper triangular matrix $U$. Since the determinant of a triangular matrix is the product of its diagonal entries,

\det(A) = (-1)^{s}\, \prod_{i=1}^{n} u_{ii}. \tag{4}

This computes $\det(A)$ in $O(n^3)$ operations — the same cost as the elimination itself.

Connection to invertibility

$A$ is invertible exactly when Gaussian elimination produces a nonzero entry on every diagonal of $U$, i.e., $\prod_i u_{ii} \ne 0$. By (4), this is equivalent to $\det(A) \ne 0$ — consistent with the criterion established in Determinant.

Multiplicativity

\det(AB) = \det(A)\,\det(B). \tag{5}

The determinant is a multiplicative homomorphism from $n \times n$ matrices to the field $F$. Several immediate consequences follow:

• $\det(I_n) = 1$, so if $A$ is invertible then $1 = \det(A)\det(A^{-1})$, giving $\det(A^{-1}) = 1/\det(A)$.
• $\det(A^k) = \det(A)^k$ for any non-negative integer $k$.
• $\det(A_1 A_2 \cdots A_k) = \det(A_1)\det(A_2)\cdots\det(A_k)$.

Note that (5) does not extend to sums: $\det(A + B) \ne \det(A) + \det(B)$ in general.

Adjugate matrix

The cofactor matrix of $A$ is the $n \times n$ matrix whose $(i,j)$ entry is the cofactor $C_{ij} = (-1)^{i+j}\det(A_{ij})$.

The adjugate (also called the classical adjoint) of $A$ is the transpose of the cofactor matrix:

\operatorname{adj}(A)_{ij} \coloneqq C_{ji} = (-1)^{i+j}\det(A_{ji}). \tag{6}

Theorem:

A \cdot \operatorname{adj}(A) = \operatorname{adj}(A) \cdot A = \det(A)\, I_n. \tag{7}

The proof follows by writing out the $(i,k)$ entry of $A \cdot \operatorname{adj}(A)$ as $\sum_j a_{ij} C_{kj}$: when $i = k$ this is the cofactor expansion of $\det(A)$ along row $i$; when $i \ne k$ it equals zero because it computes the determinant of a matrix with two identical rows.

When $\det(A) \ne 0$, dividing both sides by $\det(A)$ gives an explicit inverse formula:

A^{-1} = \frac{1}{\det(A)}\operatorname{adj}(A). \tag{8}

For large $n$ this is $O(n^4)$ — slower than Gaussian elimination. It is most useful for $2 \times 2$ and $3 \times 3$ matrices by hand, or in theoretical arguments.

The 2×2 case

For $A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$ with $\det(A) = ad - bc \ne 0$:

$A^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}.$

Swap the diagonal entries, negate the off-diagonal entries, and divide by the determinant.

Cramer’s rule

Let $A$ be an invertible $n \times n$ matrix and $b \in F^n$. The unique solution of $Ax = b$ has components

x_i = \frac{\det(A_i(b))}{\det(A)}, \qquad i = 1, \ldots, n, \tag{9}

where $A_i(b)$ is the matrix obtained from $A$ by replacing its $i$-th column with $b$.

Why it works: From $x = A^{-1}b = \frac{1}{\det(A)}\operatorname{adj}(A)\,b$, the $i$-th component is $\frac{1}{\det(A)}$ times the dot product of the $i$-th row of $\operatorname{adj}(A)$ with $b$. Expanding that dot product as a cofactor expansion along column $i$ of $A$ (with $b$ inserted) gives exactly $\det(A_i(b))$.

Like the adjugate formula, Cramer’s rule is $O(n^4)$ and is impractical for large systems. It is mainly used for $n \le 3$ by hand and in theoretical proofs (such as showing that the solution of an integer linear system has a rational — or integer — representation).

Summary

• Laplace expansion works along any row or column (equations (1) and (2)); choose the one with the most zeros to minimize work.
• Transpose: $\det(A^\top) = \det(A)$, so every row result has an exact column analogue.
• Row operations affect det predictably: swapping two rows flips the sign; scaling a row by $c$ scales det by $c$; adding a multiple of one row to another leaves det unchanged.
• Gaussian elimination gives an $O(n^3)$ algorithm: reduce $A$ to upper triangular $U$, then $\det(A) = (-1)^{\text{swaps}} \prod_i u_{ii}$.
• Multiplicativity: $\det(AB) = \det(A)\det(B)$; in particular $\det(A^{-1}) = 1/\det(A)$.
• Adjugate: $A^{-1} = \frac{1}{\det(A)}\operatorname{adj}(A)$, an explicit but $O(n^4)$ formula, mainly useful for small matrices or theoretical work.
• Cramer’s rule: $x_i = \det(A_i(b))/\det(A)$, a closed-form solution for $Ax = b$, practical for $n \le 3$ or theoretical arguments.