Solved problems: Calculating determinants of 2x2 and 3x3 matrices

📚 Introduction to Determinants

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix. The determinant provides important information about the matrix, such as whether the matrix has an inverse and the volume scaling factor of the linear transformation described by the matrix. Let's explore how to calculate determinants for 2x2 and 3x3 matrices.

📜 History and Background

The concept of determinants arose independently in different parts of the world. In Japan, Seki Takakazu used determinants in the 17th century, while Gottfried Wilhelm Leibniz also employed them around the same time. However, it was Augustin-Louis Cauchy who, in the 19th century, formalized the theory of determinants as we know it today.

➗ Calculating the Determinant of a 2x2 Matrix

For a 2x2 matrix, the determinant is calculated using a simple formula. Given a matrix:

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

The determinant, denoted as det(A) or |A|, is:

$det(A) = ad - bc$

• ➕ Step 1: Multiply the elements on the main diagonal (top-left to bottom-right).
• ➖ Step 2: Multiply the elements on the off-diagonal (top-right to bottom-left).
• 🧮 Step 3: Subtract the result of Step 2 from the result of Step 1.

🧪 Example 1: 2x2 Matrix

Let's calculate the determinant of the following matrix:

$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$

det(A) = (2 * 4) - (3 * 1) = 8 - 3 = 5

📐 Calculating the Determinant of a 3x3 Matrix

For a 3x3 matrix, the determinant can be calculated using the rule of Sarrus or cofactor expansion. We'll demonstrate the rule of Sarrus, as it's generally easier to remember.

Given a matrix:

$B = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$

The determinant is:

$det(B) = a(ei - fh) - b(di - fg) + c(dh - eg)$

Or, using the Rule of Sarrus, rewrite the first two columns next to the matrix and sum the products of the diagonals:

• ✍️ Step 1: Rewrite the first two columns of the matrix next to it.
• ➕ Step 2: Sum the products of the three diagonals going from top-left to bottom-right.
• ➖ Step 3: Sum the products of the three diagonals going from top-right to bottom-left.
• ➗ Step 4: Subtract the sum from Step 3 from the sum in Step 2.

📊 Example 2: 3x3 Matrix

Let's calculate the determinant of the following matrix:

$B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

det(B) = 1(5*9 - 6*8) - 2(4*9 - 6*7) + 3(4*8 - 5*7) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = -3 - 2(-6) + 3(-3) = -3 + 12 - 9 = 0

💡 Real-world Applications

• 🌍 Linear Transformations: Determinants reveal how a linear transformation scales areas or volumes.
• 💻 Computer Graphics: Used in transformations, projections, and determining if points are coplanar.
• ⚙️ Engineering: Solving systems of linear equations in structural analysis and circuit design.

✅ Practice Quiz

Calculate the determinants of the following matrices:

1. 🔢 Matrix 1: $\begin{bmatrix} 5 & 2 \\ 1 & 3 \end{bmatrix}$
2. 🔢 Matrix 2: $\begin{bmatrix} -1 & 4 \\ 2 & -3 \end{bmatrix}$
3. 🔢 Matrix 3: $\begin{bmatrix} 0 & 5 \\ -2 & 1 \end{bmatrix}$
4. 🔢 Matrix 4: $\begin{bmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 1 & 2 & 1 \end{bmatrix}$
5. 🔢 Matrix 5: $\begin{bmatrix} 2 & -1 & 0 \\ 1 & 1 & -1 \\ 0 & 1 & 2 \end{bmatrix}$
6. 🔢 Matrix 6: $\begin{bmatrix} 3 & 1 & 2 \\ 0 & 2 & 5 \\ 0 & 0 & -4 \end{bmatrix}$
7. 🔢 Matrix 7: $\begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$

🔑 Conclusion

Calculating determinants is a fundamental skill in linear algebra. Understanding how to compute them for 2x2 and 3x3 matrices is essential for various applications in mathematics, science, and engineering. With practice, you'll master this important concept!