Mathematics Lecture Notes

Definition4.1Determinant (recursive)

The determinant $\det : M_{n \times n}(F) \to F$ is defined recursively:

Base case ( $n = 1$ ): $\det(a) = a$ .

Recursive step: For $n \geq 2$ , expanding along the first row:

$\det(A) = \sum_{j=1}^n (-1)^{1+j} a_{1j} \det(A_{1j})$

where $A_{1j}$ is the $(n-1) \times (n-1)$ matrix obtained by deleting row 1 and column $j$ .

Definition4.2Determinant via permutations (Leibniz formula)

Equivalently,

$\det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}$

where $S_n$ is the symmetric group on $\{1, \ldots, n\}$ and $\text{sgn}(\sigma) = \pm 1$ is the sign of the permutation.

Example1 x 1 determinant

$\det(a) = a$ .

Example2 x 2 determinant

$\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc.$

This is the area of the parallelogram spanned by the column vectors $(a, c)$ and $(b, d)$ , with sign indicating orientation.

Example3 x 3 determinant (Sarrus' rule)

$\det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = aei + bfg + cdh - ceg - bdi - afh.$

This is the signed volume of the parallelepiped spanned by the three column vectors.

ExampleDeterminant of the identity

$\det(I_n) = 1$ for all $n$ . (The identity preserves volume and orientation.)

ExampleDeterminant of a diagonal matrix

$\det(\text{diag}(d_1, \ldots, d_n)) = d_1 d_2 \cdots d_n$ . The determinant of a diagonal matrix is the product of the diagonal entries.

ExampleDeterminant of a triangular matrix

For upper or lower triangular matrices, the determinant equals the product of the diagonal entries:

$\det\begin{pmatrix} 2 & 3 & 1 \\ 0 & -1 & 4 \\ 0 & 0 & 5 \end{pmatrix} = 2 \cdot (-1) \cdot 5 = -10.$

ExampleZero row or column

If $A$ has a row (or column) of all zeros, then $\det(A) = 0$ . (In the permutation formula, every term contains a factor $a_{i,\sigma(i)} = 0$ from the zero row.)

Theorem4.1Properties of the determinant

For $A, B \in M_{n \times n}(F)$ and $c \in F$ :

Multiplicativity: $\det(AB) = \det(A) \cdot \det(B)$ .
Transpose: $\det(A^T) = \det(A)$ .
Scalar multiple: $\det(cA) = c^n \det(A)$ .
Inverse: If $A$ is invertible, $\det(A^{-1}) = (\det A)^{-1}$ .
Row operations:
- Row swap: $\det$ changes sign.
- Scale row by $c$ : $\det$ multiplies by $c$ .
- Add multiple of one row to another: $\det$ unchanged.

ExampleMultiplicativity

$A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ , $B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ .

$\det(A) = 6$ , $\det(B) = 1$ . $AB = \begin{pmatrix} 2 & 2 \\ 0 & 3 \end{pmatrix}$ , $\det(AB) = 6 = 6 \cdot 1$ .

ExampleScalar multiple

$\det(2I_3) = 2^3 = 8$ , not $2$ . This is a common mistake: scaling an $n \times n$ matrix by $c$ multiplies the determinant by $c^n$ , not $c$ .

ExampleDeterminant is NOT additive

In general, $\det(A + B) \neq \det(A) + \det(B)$ .

$A = B = I_2$ : $\det(A + B) = \det(2I_2) = 4 \neq 2 = 1 + 1$ .

ExampleSimilar matrices have the same determinant

If $B = P^{-1}AP$ , then $\det(B) = \det(P^{-1})\det(A)\det(P) = \det(A)$ . The determinant is a similarity invariant and hence an invariant of the linear operator, independent of the choice of basis.

ExampleComputing det via row operations

$\det\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 10 \end{pmatrix}$ :

$R_2 \to R_2 - 4R_1$ , $R_3 \to R_3 - 7R_1$ : $\begin{pmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & -6 & -11 \end{pmatrix}$ .

$R_3 \to R_3 - 2R_2$ : $\begin{pmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \\ 0 & 0 & 1 \end{pmatrix}$ .

$\det = 1 \cdot (-3) \cdot 1 = -3$ (product of pivots, no row swaps so sign is positive).

RemarkVolume and orientation

For $A \in M_{n \times n}(\mathbb{R})$ with columns $v_1, \ldots, v_n$ :

$|\det(A)|$ = the $n$ -dimensional volume of the parallelepiped spanned by $v_1, \ldots, v_n$ .
$\text{sgn}(\det(A))$ = the orientation: $+1$ preserves orientation, $-1$ reverses it.
$\det(A) = 0$ iff the vectors are linearly dependent (the parallelepiped is degenerate, with zero volume).

ExampleArea of a parallelogram

Columns $(3, 0)^T$ and $(1, 2)^T$ : $\det\begin{pmatrix} 3 & 1 \\ 0 & 2 \end{pmatrix} = 6$ . The parallelogram has area $|6| = 6$ .

ExampleRotation preserves volume

$\det(R_\theta) = \cos^2\theta + \sin^2\theta = 1$ . Rotations preserve both area and orientation.

ExampleReflection reverses orientation

$\det\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = -1$ . Reflection preserves area but reverses orientation.

RemarkLooking ahead

The determinant provides the scalar test for invertibility (Theorem 4.2) and leads to Cramer's Rule for solving systems. The cofactor expansion gives a practical recursive computation method.

Math Notes

The Determinant

Definition

Small cases

Properties

Geometric interpretation