Determinants

This chapter systematically studies the definition, properties, calculation methods of determinants, and their applications in systems of linear equations.

Definition of Determinants

• Second-order determinant: $$|A| = \begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}$$
• Third-order determinant: $$|A| = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix}$$ Expanding along the first row: $$= a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} - a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{vmatrix}$$
• n-th order determinant: recursively expand along a row or column

Properties of Determinants

1. Swapping two rows (or columns) changes the sign of the determinant
2. If a row (or column) is all zeros, the determinant is zero
3. A common factor can be factored out from a row (or column)
4. Adding a multiple of one row (or column) to another does not change the determinant
5. The determinant is linear in each row (or column)
6. The determinant of a diagonal matrix is the product of the diagonal elements
7. The determinant of an upper (or lower) triangular matrix is the product of the diagonal elements
8. The determinant is unchanged by transposition

Methods for Calculating Determinants

• Expansion by row or column
• Simplification using properties
• Block determinants

Determinants and Systems of Linear Equations

• Cramer’s rule: $Ax = b$ has a unique solution if and only if $|A| \neq 0$

---

Exercises

1. Calculate the second-order determinant $\begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix}$.
2. Calculate the third-order determinant $\begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix}$.
3. Determine whether the determinant $\begin{vmatrix} 1 & 2 \\ 2 & 4 \end{vmatrix}$ is zero and explain why.
4. Use the properties of determinants to simplify the calculation of $\begin{vmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{vmatrix}$.
5. Let $A$ be a $3 \times 3$ matrix with $|A| = 2$. If the first and second rows of $A$ are swapped, what is the value of the new determinant?

Reference Answers

1. Calculate the second-order determinant $\begin{vmatrix} 2 & 3 \\ 1 & 4 \end{vmatrix}$

$2 \times 4 - 3 \times 1 = 8 - 3 = 5$

---

2. Calculate the third-order determinant $\begin{vmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{vmatrix}$

Expanding along the first row: $1 \begin{vmatrix} 1 & 4 \\ 6 & 0 \end{vmatrix} - 2 \begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} + 3 \begin{vmatrix} 0 & 1 \\ 5 & 6 \end{vmatrix}$ $= 1(1 \times 0 - 4 \times 6) - 2(0 \times 0 - 4 \times 5) + 3(0 \times 6 - 1 \times 5)$ $= 1(0 - 24) - 2(0 - 20) + 3(0 - 5)$ $= -24 + 40 - 15 = 1$

---

3. Determine whether $\begin{vmatrix} 1 & 2 \\ 2 & 4 \end{vmatrix}$ is zero and explain why

$1 \times 4 - 2 \times 2 = 4 - 4 = 0$, the two rows are proportional, so the determinant is zero.

---

4. Use the properties of determinants to simplify $\begin{vmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{vmatrix}$

All three rows are proportional, so the determinant is zero.

---

5. Let $A$ be a $3 \times 3$ matrix with $|A| = 2$. If the first and second rows of $A$ are swapped, what is the value of the new determinant?

The sign changes: $-2$

---