🌟 Level 3 - Topic 11: Matrix Algebra (Part 3) - Determinants & Inverses (2x2) 🔍🔄

1) Introduction to Determinants and Inverses (2x2) 🔍🔄

Welcome back! In Part 2, we mastered matrix multiplication. Now, in Part 3 of Matrix Algebra, we're going to explore two more essential concepts for square matrices, especially focusing on \(2 \times 2\) matrices: Determinants and Inverses. These concepts are crucial for understanding matrix properties and solving systems of linear equations using matrices. The determinant is a special number that can be calculated from a square matrix, offering insights into the matrix's properties. The inverse of a matrix, similar to the reciprocal of a number, allows us to 'divide' in matrix algebra, in a sense. In this part, we will focus on \(2 \times 2\) matrices to build a solid foundation.

For a square matrix, the Determinant is a special number that can be calculated from its elements. For a \(2 \times 2\) matrix, it's a simple calculation. The Inverse of a square matrix (if it exists) is another matrix that, when multiplied by the original matrix, gives the Identity Matrix.

2) Determinant of a 2x2 Matrix 🔍

The determinant is a value that can be computed from the elements of a square matrix. For a \(2 \times 2\) matrix, the determinant is quite straightforward to calculate.

2.1 Calculating the Determinant

For a \(2 \times 2\) matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the determinant is defined as follows:

The determinant of a \(2 \times 2\) matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), denoted as \( \det(A) \) or \( |A| \), is given by: \( \det(A) = ad - bc \).

It's the product of the main diagonal elements (\(a\) and \(d\)) minus the product of the off-diagonal elements (\(b\) and \(c\)).

Example 1: Determinant of a 2x2 Matrix

Calculate the determinant of matrix \( A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \).

\( \det(A) = (3 \times 4) - (1 \times 2) = 12 - 2 = 10 \).

So, the determinant of matrix \(A\) is 10.

Example 2: Determinant with Negative Elements

Calculate the determinant of matrix \( B = \begin{bmatrix} -2 & 3 \\ 5 & -1 \end{bmatrix} \).

\( \det(B) = ((-2) \times (-1)) - (3 \times 5) = 2 - 15 = -13 \).

The determinant of matrix \(B\) is -13.

2.2 Significance of the Determinant

The determinant provides important information about a matrix. For \(2 \times 2\) matrices (and square matrices in general):

A non-zero determinant (like in Example 1 and 2) indicates that the matrix is invertible, meaning it has an inverse.
A zero determinant means the matrix is not invertible, or singular, and it does not have an inverse.

We will explore matrix inverses in the next section.

3) Inverse of a 2x2 Matrix 🔄

The inverse of a matrix is another matrix that, when multiplied with the original matrix, results in the Identity Matrix. Not all matrices have inverses. For \(2 \times 2\) matrices, the determinant plays a crucial role in determining if an inverse exists, and in calculating it.

3.1 Condition for Inverse and Formula

A \(2 \times 2\) matrix \( A \) is invertible (has an inverse) if and only if its determinant \( \det(A) \neq 0 \). If \( \det(A) \neq 0 \), the inverse of \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), denoted as \( A^{-1} \), is given by: \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).

If \( \det(A) = 0 \), then matrix \(A\) is singular and has no inverse.

To find the inverse of a \(2 \times 2\) matrix:

Calculate the determinant \( \det(A) = ad - bc \).
If \( \det(A) = 0 \), the inverse does not exist.
If \( \det(A) \neq 0 \), swap the positions of \(a\) and \(d\), change the signs of \(b\) and \(c\), and multiply the resulting matrix by \( \frac{1}{\det(A)} \).

Example 3: Finding the Inverse of a 2x2 Matrix

Find the inverse of matrix \( A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \).

Step 1: Calculate the determinant.

From Example 1, \( \det(A) = 10 \). Since \( \det(A) \neq 0 \), the inverse exists.

Step 2: Apply the inverse formula.

\( A^{-1} = \frac{1}{10} \begin{bmatrix} 4 & -1 \\ -2 & 3 \end{bmatrix} = \begin{bmatrix} \frac{4}{10} & \frac{-1}{10} \\ \frac{-2}{10} & \frac{3}{10} \end{bmatrix} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix} \)

So, \( A^{-1} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix} \).

Example 4: Matrix with No Inverse

Determine if matrix \( S = \begin{bmatrix} 2 & 4 \\ 1 & 2 \end{bmatrix} \) has an inverse.

Step 1: Calculate the determinant.

\( \det(S) = (2 \times 2) - (4 \times 1) = 4 - 4 = 0 \).

Since \( \det(S) = 0 \), matrix \(S\) is singular and does not have an inverse.

3.2 Verifying the Inverse

To verify that \(A^{-1}\) is indeed the inverse of \(A\), we should check if \( A \times A^{-1} = I \) and \( A^{-1} \times A = I \), where \(I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) is the \(2 \times 2\) Identity Matrix.

Example 5: Verifying the Inverse

Let's verify if \( A^{-1} = \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix} \) is the inverse of \( A = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \).

Check \( A \times A^{-1} \).

\( A \times A^{-1} = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} \frac{2}{5} & -\frac{1}{10} \\ -\frac{1}{5} & \frac{3}{10} \end{bmatrix} = \begin{bmatrix} (3\times \frac{2}{5} + 1\times (-\frac{1}{5})) & (3\times (-\frac{1}{10}) + 1\times \frac{3}{10}) \\ (2\times \frac{2}{5} + 4\times (-\frac{1}{5})) & (2\times (-\frac{1}{10}) + 4\times \frac{3}{10}) \end{bmatrix} \)

\( = \begin{bmatrix} (\frac{6}{5} - \frac{1}{5}) & (-\frac{3}{10} + \frac{3}{10}) \\ (\frac{4}{5} - \frac{4}{5}) & (-\frac{2}{10} + \frac{12}{10}) \end{bmatrix} = \begin{bmatrix} \frac{5}{5} & 0 \\ 0 & \frac{10}{10} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I_2 \)

Similarly, you can check that \( A^{-1} \times A = I_2 \) as well (try it as an exercise!). Since \( A \times A^{-1} = A^{-1} \times A = I_2 \), \(A^{-1}\) is indeed the inverse of \(A\).

4) Practice Questions 🎯

5.1 Fundamental – Build Skills

1. Calculate the determinant of matrix \(A = \begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix}\).

2. Calculate the determinant of matrix \(B = \begin{bmatrix} -1 & 5 \\ 2 & -3 \end{bmatrix}\).

3. Calculate the determinant of matrix \(C = \begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix}\).

4. Is matrix \(A = \begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix}\) invertible? (Use your result from question 1).

5. Is matrix \(C = \begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix}\) invertible? (Use your result from question 3).

6. Find the inverse of matrix \(A = \begin{bmatrix} 4 & 2 \\ 3 & 1 \end{bmatrix}\) (if it exists).

7. Find the inverse of matrix \(D = \begin{bmatrix} -2 & 1 \\ 3 & 0 \end{bmatrix}\) (if it exists).

8. Verify that your answer for the inverse of matrix \(A\) (from question 6) is correct by multiplying \(A\) and \(A^{-1}\).

9. For what value(s) of \(k\) is the matrix \(M = \begin{bmatrix} k & 2 \\ 3 & 4 \end{bmatrix}\) singular (not invertible)?

10. Given \(A = \begin{bmatrix} 2 & 2 \\ 2 & 3 \end{bmatrix}\), calculate \( \det(A) \) and \( A^{-1} \).

5.2 Challenging – Push Limits 💪🚀

1. If \( \det(X) = -5 \) for matrix \(X = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), find \( \det(2X) \). (Hint: How does scalar multiplication affect the determinant?).

2. For matrices \(A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\), calculate \( \det(AB) \) and compare it with \( \det(A) \times \det(B) \). What do you observe?

3. Find the inverse of \(A = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}\). (This type of matrix is important in rotations).

4. (Conceptual) Explain why a matrix with a determinant of zero does not have an inverse. What does a zero determinant tell you about the transformation represented by the matrix?

5. (Application) A system of linear equations is given in matrix form as \(AX = B\), where \(A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 8 \\ 13 \end{bmatrix}\). Use the inverse of matrix \(A\) to solve for \(X = \begin{bmatrix} x \\ y \end{bmatrix}\). (Recall that \(X = A^{-1}B\)).

6) Summary 🎉

Determinant of a 2x2 Matrix: For \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), \( \det(A) = ad - bc \).
Invertibility: A \(2 \times 2\) matrix \(A\) is invertible if \( \det(A) \neq 0 \).
Inverse Formula: If \( \det(A) \neq 0 \), then \( A^{-1} = \frac{1}{\det(A)} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \).
Verification: To check if \(B\) is the inverse of \(A\), verify if \( AB = BA = I \).

Congratulations! You've now explored determinants and inverses of \(2 \times 2\) matrices, adding powerful tools to your matrix toolkit. These concepts are foundational for more advanced topics in linear algebra and have applications in various fields. In future topics, we might extend these ideas to larger matrices and explore more of their applications. Keep practicing, and you'll strengthen your matrix skills! 🌟