Multiplicativity and Invertibility

The relationship between determinants, matrix multiplication, and invertibility forms the cornerstone of determinant theory. These properties make the determinant an invaluable computational and theoretical tool.

TheoremMultiplicativity of Determinants

For any $n \times n$ matrices $A$ and $B$ : $\det(AB) = \det(A) \cdot \det(B)$

This property extends to products of any number of matrices: $\det(A_1A_2 \cdots A_k) = \det(A_1)\det(A_2) \cdots \det(A_k)$ .

Multiplicativity implies that the determinant is a group homomorphism from $GL_n(\mathbb{F})$ (invertible matrices) to $\mathbb{F}^*$ (nonzero scalars). This algebraic perspective unifies many determinant properties.

TheoremDeterminant and Invertibility

An $n \times n$ matrix $A$ is invertible if and only if $\det(A) \neq 0$ .

Moreover, if $A$ is invertible: $\det(A^{-1}) = \frac{1}{\det(A)} = [\det(A)]^{-1}$

Proof: If $A$ is invertible, then $AA^{-1} = I$ . Taking determinants: $\det(A)\det(A^{-1}) = \det(I) = 1$

Thus $\det(A) \neq 0$ and $\det(A^{-1}) = 1/\det(A)$ .

Conversely, if $\det(A) \neq 0$ , the inverse formula $A^{-1} = \frac{1}{\det(A)}\text{adj}(A)$ shows $A$ is invertible.

TheoremProperties Under Matrix Operations

Let $A$ be an $n \times n$ matrix.

Transpose: $\det(A^T) = \det(A)$
Scalar multiple: $\det(cA) = c^n \det(A)$ for scalar $c$
Powers: $\det(A^k) = [\det(A)]^k$ for positive integer $k$
Similar matrices: If $B = P^{-1}AP$ , then $\det(B) = \det(A)$
Block diagonal: If $A = \begin{bmatrix} B & 0 \\ 0 & C \end{bmatrix}$ , then $\det(A) = \det(B)\det(C)$

ExampleUsing Multiplicativity

Suppose $\det(A) = 2$ and $\det(B) = -3$ for $3 \times 3$ matrices. Find $\det(A^2B^{-1})$ .

$\det(A^2B^{-1}) = \det(A^2)\det(B^{-1}) = [\det(A)]^2 \cdot \frac{1}{\det(B)} = 4 \cdot \frac{1}{-3} = -\frac{4}{3}$

TheoremDeterminant of Elementary Matrices

Each elementary row operation corresponds to multiplication by an elementary matrix:

Row swap: $\det(E) = -1$
Row scaling by $c$ : $\det(E) = c$
Row addition: $\det(E) = 1$

Since any invertible matrix is a product of elementary matrices, $\det(A) = \det(E_1 \cdots E_k) = \det(E_1) \cdots \det(E_k)$ .

Remark

The multiplicativity property $\det(AB) = \det(A)\det(B)$ is remarkable: matrix multiplication is complicated, yet the determinant converts it to simple scalar multiplication. This reduction of complexity is what makes determinants powerful in applications from eigenvalue theory to differential equations.