Step-by-Step Solutions for 3x3 Determinants: Pre-Calculus Examples

📚 Quick Study Guide

• 🔢 A determinant is a scalar value that can be computed from the elements of a square matrix.
• 📐 For a 3x3 matrix, the determinant can be found using expansion by minors (cofactor expansion).
• ➕ The determinant of a 3x3 matrix $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ is calculated as: $det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$.
• 💡 Sarrus' rule is a mnemonic device to remember the formula: write out the first two columns again to the right of the matrix, then sum the products of the diagonals going from top-left to bottom-right and subtract the products of the diagonals going from top-right to bottom-left.

Practice Quiz

1. Question 1: What is the determinant of the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$?
   1. -26
   2. 26
   3. 13
   4. -13
2. Question 2: Calculate the determinant of the matrix $\begin{bmatrix} 2 & 0 & 1 \\ 3 & 1 & 2 \\ 1 & 0 & 3 \end{bmatrix}$.
   1. 5
   2. 10
   3. -5
   4. -10
3. Question 3: Find the determinant of $\begin{bmatrix} 1 & -1 & 2 \\ 2 & 1 & -1 \\ 3 & 0 & 1 \end{bmatrix}$.
   1. 9
   2. -9
   3. 0
   4. 1
4. Question 4: What is the determinant of the matrix $\begin{bmatrix} 4 & 2 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 5 \end{bmatrix}$?
   1. 60
   2. 30
   3. 0
   4. 1
5. Question 5: Evaluate the determinant of $\begin{bmatrix} -1 & 0 & 2 \\ 3 & 1 & 4 \\ 0 & 2 & -1 \end{bmatrix}$.
   1. 15
   2. -15
   3. -9
   4. 9
6. Question 6: Determine the determinant of the matrix $\begin{bmatrix} 2 & 1 & 3 \\ 5 & 0 & 1 \\ 1 & 2 & 0 \end{bmatrix}$.
   1. -29
   2. 29
   3. -15
   4. 15
7. Question 7: What is the determinant of the matrix $\begin{bmatrix} 1 & 2 & 1 \\ 0 & 1 & 0 \\ 2 & 3 & 2 \end{bmatrix}$?
   1. 0
   2. 1
   3. 2
   4. -1