Determinant of a Matrix

• The determinant is a value computed from the elements of a square matrix.

Definition

• For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is $det(A) = ad - bc$.

Example

• For $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$, $det(A) = (1)(4) - (2)(3) = 4 - 6 = -2$.

Definition

• For an $n \times n$ matrix A, the $(i,j)$-minor, denoted $M_{ij}$, is the determinant of the $(n-1) \times (n-1)$ matrix formed by removing the $i^{th}$ row and $j^{th}$ column from A.

Example

• For $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$, $M_{11} = det \begin{bmatrix} 5 & 6 \ 8 & 9 \end{bmatrix} = (5)(9) - (6)(8) = 45 - 48 = -3$.

Definition

• For an $n \times n$ matrix A, the $(i,j)$-cofactor, denoted $C_{ij}$, is defined as $C_{ij} = (-1)^{i+j} M_{ij}$.

Example

• For $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$, $C_{11} = (-1)^{1+1} M_{11} = (1)(-3) = -3$.

Definition

• The determinant of an $n \times n$ matrix A can be calculated by cofactor expansion along the first row: $det(A) = a_{11}C_{11} + a_{12}C_{12} +... + a_{1n}C_{1n}$.

Theorem 5.1.1

• The determinant of A can be computed by cofactor expansion along any row $i$: $det(A) = a_{i1}C_{i1} + a_{i2}C_{i2} +... + a_{in}C_{in}$.
• The determinant of A can be computed by cofactor expansion down any column $j$: $det(A) = a_{1j}C_{1j} + a_{2j}C_{2j} +... + a_{nj}C_{nj}$.

Example

• Compute the determinant of $A = \begin{bmatrix} 1 & 5 & 0 \ 2 & 4 & -1 \ 0 & -2 & 0 \end{bmatrix}$ by cofactor expansion along the third row.
• $det(A) = a_{31}C_{31} + a_{32}C_{32} + a_{33}C_{33} = (0)C_{31} + (-2)C_{32} + (0)C_{33} = -2C_{32}$.
• $C_{32} = (-1)^{3+2} M_{32} = (-1)^5 det \begin{bmatrix} 1 & 0 \ 2 & -1 \end{bmatrix} = (-1)((1)(-1) - (0)(2)) = (-1)(-1) = 1$.
• $det(A) = -2 \cdot 1 = -2$.

Theorem 5.1.2

• If A is a triangular matrix, $det(A)$ is the product of the entries on the main diagonal of A.

Example

• For $A = \begin{bmatrix} 1 & 2 & 3 \ 0 & 4 & 5 \ 0 & 0 & 6 \end{bmatrix}$, $det(A) = (1)(4)(6) = 24$.