Properties of Determinants

Overview

Determinants have many useful properties that simplify calculations and provide insights into matrix behavior. Understanding these properties is essential for efficient computation.

Main Properties

1. Determinant of Transpose

$\det(A^T) = \det(A)$

Row and column operations have the same effect on the determinant.

2. Multiplicative Property

$\det(AB) = \det(A) \times \det(B)$

3. Determinant of Inverse

$\det(A^{-1}) = \frac{1}{\det(A)}$

(Only when $A$ is invertible)

4. Scalar Multiplication

For an $n \times n$ matrix $A$ and scalar $c$:

$\det(cA) = c^n \det(A)$

Effect of Row Operations

Operation	Effect on $\det(A)$
Swap two rows	$\det$ changes sign
Multiply row by $c$	$\det$ multiplied by $c$
Add multiple of one row to another	$\det$ unchanged

Examples

If $\det(A) = 5$:

• Swap $R_1 \leftrightarrow R_2$: $\det = -5$
• Scale $R_1$ by 3: $\det = 15$
• $R_2 + 4R_1 \to R_2$: $\det = 5$ (unchanged)

Special Matrix Determinants

Triangular Matrices

For triangular matrices (upper or lower):

$\det(A) = a_{11} \times a_{22} \times \cdots \times a_{nn}$

Product of diagonal entries.

Diagonal Matrices

$\det(D) = d_1 \times d_2 \times \cdots \times d_n$

Identity Matrix

$\det(I) = 1$

Zero Matrix

$\det(O) = 0$

Determinant and Invertibility

The following are equivalent:

• $\det(A) \neq 0$
• $A$ is invertible
• $A$ has full rank
• $A\mathbf{x} = \mathbf{0}$ has only the trivial solution
• Columns of $A$ are linearly independent

Properties List

Property	Formula
Transpose	$\det(A^T) = \det(A)$
Product	$\det(AB) = \det(A)\det(B)$
Inverse	$\det(A^{-1}) = 1/\det(A)$
Scalar	$\det(cA) = c^n\det(A)$
Power	$\det(A^n) = (\det(A))^n$
Similar	$\det(P^{-1}AP) = \det(A)$

Zero Determinant Conditions

$\det(A) = 0$ if:

• $A$ has a row of zeros
• $A$ has a column of zeros
• Two rows are identical
• Two columns are identical
• A row is a multiple of another row
• A column is a multiple of another column

Computing Determinants Efficiently

Strategy 1: Row Reduce to Triangular Form

1. Use row operations (tracking sign changes)
2. Compute product of diagonal entries

Strategy 2: Expansion Along Sparse Row/Column

Choose the row or column with the most zeros for cofactor expansion.

Examples

Example 1: Product Rule

$\det(A) = 2, \quad \det(B) = 3$

$\det(AB) = 2 \times 3 = 6$

$\det(A^2) = 2^2 = 4$

$\det(A^{-1}) = \frac{1}{2}$

Example 2: Row Operation

$A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix} \quad \det(A) = 2(8) - 4(6) = -8$

Swap rows:

$B = \begin{bmatrix} 6 & 8 \\ 2 & 4 \end{bmatrix} \quad \det(B) = 6(4) - 8(2) = 8 = -\det(A) \checkmark$

Example 3: Scalar Multiplication

$A$ is $3 \times 3$ with $\det(A) = 5$

$\det(2A) = 2^3 \times 5 = 40$