🌟 Level 3 - Topic 12: Matrix Algebra (Part 4) - Determinants & Inverses (3x3) 🔍⏫

1) Introduction to Determinants and Inverses (3x3) 🔍⏫

Welcome back to Matrix Algebra! In Part 3, we explored determinants and inverses for \(2 \times 2\) matrices. Now, in Part 4, we are ready to scale up to 3x3 Matrices! We will learn how to calculate determinants and inverses of \(3 \times 3\) matrices, which are more complex but follow logical extensions of the concepts we've already learned. Understanding \(3 \times 3\) matrices is crucial as they are widely used in 3D graphics, engineering, and advanced mathematical models.

Extending our understanding from \(2 \times 2\) matrices, for a square matrix, the Determinant is a value with specific calculation methods that provide key information. For a \(3 \times 3\) matrix, the calculation is more involved than for \(2 \times 2\), but still manageable. The Inverse of a \(3 \times 3\) matrix, when it exists, is also a \(3 \times 3\) matrix that acts as the 'reciprocal' in matrix multiplication.

In this topic, we will cover:

  • Calculating the Determinant of a 3x3 Matrix using cofactor expansion.
  • Determining if a \(3 \times 3\) matrix is invertible using its determinant.
  • Finding the Inverse of a 3x3 Matrix using the adjugate formula.
  • Working through examples and practice problems to solidify your skills.

Let's tackle the world of \(3 \times 3\) matrices and expand our matrix algebra abilities! 🚀

2) Determinant of a 3x3 Matrix 🔍

Calculating the determinant of a \(3 \times 3\) matrix is more complex than for a \(2 \times 2\) matrix. We will use a method called cofactor expansion (or expansion by minors).

2.1 Cofactor Expansion Method

For a \(3 \times 3\) matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \), the determinant can be calculated by expanding along any row or column. Let's expand along the first row. The formula is:

For a \(3 \times 3\) matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \), the determinant \( \det(A) \) is: \( \det(A) = a \cdot C_{11} + b \cdot C_{12} + c \cdot C_{13} \)

Where \( C_{ij} \) is the cofactor of the element in the \(i^{th}\) row and \(j^{th}\) column. The cofactor \( C_{ij} \) is calculated as \( C_{ij} = (-1)^{i+j} M_{ij} \), and \( M_{ij} \) is the determinant of the \(2 \times 2\) matrix obtained by deleting the \(i^{th}\) row and \(j^{th}\) column of \(A\). \(M_{ij}\) is called the minor.

Let's break down the cofactors for the first row expansion of a \(3 \times 3\) matrix:

  • For element \(a\) (position \(1,1\)): \( C_{11} = (-1)^{1+1} M_{11} = (+1) \cdot \det \begin{bmatrix} e & f \\ h & i \end{bmatrix} = (ei - fh) \)
  • For element \(b\) (position \(1,2\)): \( C_{12} = (-1)^{1+2} M_{12} = (-1) \cdot \det \begin{bmatrix} d & f \\ g & i \end{bmatrix} = -(di - fg) \)
  • For element \(c\) (position \(1,3\)): \( C_{13} = (-1)^{1+3} M_{13} = (+1) \cdot \det \begin{bmatrix} d & e \\ g & h \end{bmatrix} = (dh - eg) \)

So, expanding along the first row, the determinant of a \(3 \times 3\) matrix \(A\) is:

\( \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \)

Example 1: Determinant of a 3x3 Matrix

Calculate the determinant of matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \).

**Solution:** We'll use cofactor expansion along the first row.

  • \( C_{11} = \det \begin{bmatrix} 5 & 6 \\ 8 & 9 \end{bmatrix} = (5 \times 9) - (6 \times 8) = 45 - 48 = -3 \)
  • \( C_{12} = - \det \begin{bmatrix} 4 & 6 \\ 7 & 9 \end{bmatrix} = - ((4 \times 9) - (6 \times 7)) = - (36 - 42) = - (-6) = 6 \)
  • \( C_{13} = \det \begin{bmatrix} 4 & 5 \\ 7 & 8 \end{bmatrix} = (4 \times 8) - (5 \times 7) = 32 - 35 = -3 \)

Now, using the determinant formula: \( \det(A) = a \cdot C_{11} + b \cdot C_{12} + c \cdot C_{13} \)

\( \det(A) = (1) \cdot (-3) + (2) \cdot (6) + (3) \cdot (-3) = -3 + 12 - 9 = 0 \)

The determinant of matrix \(A\) is 0.

Example 2: Determinant with Negative Elements

Calculate the determinant of matrix \( B = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 0 & -2 \\ -2 & 1 & 1 \end{bmatrix} \).

**Solution:** Expand along the first row.

  • \( C_{11} = \det \begin{bmatrix} 0 & -2 \\ 1 & 1 \end{bmatrix} = (0 \times 1) - ((-2) \times 1) = 0 - (-2) = 2 \)
  • \( C_{12} = - \det \begin{bmatrix} 3 & -2 \\ -2 & 1 \end{bmatrix} = - ((3 \times 1) - ((-2) \times (-2))) = - (3 - 4) = - (-1) = 1 \)
  • \( C_{13} = \det \begin{bmatrix} 3 & 0 \\ -2 & 1 \end{bmatrix} = (3 \times 1) - (0 \times (-2)) = 3 - 0 = 3 \)

Determinant calculation: \( \det(B) = a \cdot C_{11} + b \cdot C_{12} + c \cdot C_{13} \)

\( \det(B) = (2) \cdot (2) + (-1) \cdot (1) + (0) \cdot (3) = 4 - 1 + 0 = 3 \)

The determinant of matrix \(B\) is 3.

2.2 Choosing Rows or Columns for Expansion

You can expand along any row or column. The formula for expansion along the \(j^{th}\) column is: \( \det(A) = \sum_{i=1}^{3} a_{ij} C_{ij} = a_{1j} C_{1j} + a_{2j} C_{2j} + a_{3j} C_{3j} \), and similarly for row expansion. When calculating by hand, it's often strategic to choose a row or column with more zeros to simplify calculations, as it reduces the number of terms you need to compute.

For instance, in Example 2, expanding along the first row was natural. If we had chosen the first row of matrix A in Example 1, it was also fine. However, if a matrix has a row or column with many zeros, expanding along that row or column simplifies the calculation significantly.

3) Inverse of a 3x3 Matrix ⏫

Similar to \(2 \times 2\) matrices, the inverse of a \(3 \times 3\) matrix \(A\), denoted as \(A^{-1}\), exists if and only if its determinant \( \det(A) \neq 0 \). The formula for the inverse of a \(3 \times 3\) matrix is more complex than for \(2 \times 2\), and it involves the adjugate (or adjoint) of the matrix and the determinant.

3.1 Adjugate of a Matrix

The adjugate (adj) of a matrix \(A\) is the transpose of the matrix of cofactors of \(A\). Let \(C\) be the matrix of cofactors of \(A\), where \(C_{ij}\) is the cofactor of \(a_{ij}\). Then, \( \text{adj}(A) = C^T \).

For a \(3 \times 3\) matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \), the matrix of cofactors is: \( C = \begin{bmatrix} C_{11} & C_{12} & C_{13} \\ C_{21} & C_{22} & C_{23} \\ C_{31} & C_{32} & C_{33} \end{bmatrix} \), and the adjugate is the transpose of \(C\): \( \text{adj}(A) = C^T = \begin{bmatrix} C_{11} & C_{21} & C_{31} \\ C_{12} & C_{22} & C_{32} \\ C_{13} & C_{23} & C_{33} \end{bmatrix} \).

Note that in the adjugate, the cofactor \(C_{ij}\) is placed in the \((j, i)\) position (transposed position).

3.2 Formula for the Inverse of a 3x3 Matrix

If \( \det(A) \neq 0 \), the inverse of a \(3 \times 3\) matrix \(A\) is given by: \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \)

To find the inverse of a \(3 \times 3\) matrix \(A\):

  1. Calculate the determinant \( \det(A) \). If \( \det(A) = 0 \), the inverse does not exist.
  2. Calculate all cofactors \( C_{ij} \) for \(i, j = 1, 2, 3\).
  3. Form the matrix of cofactors \(C\).
  4. Find the adjugate of \(A\) by transposing \(C\) to get \( \text{adj}(A) = C^T \).
  5. Multiply the adjugate by \( \frac{1}{\det(A)} \) to get \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \).

Example 3: Finding the Inverse of a 3x3 Matrix

Find the inverse of matrix \( B = \begin{bmatrix} 2 & -1 & 0 \\ 3 & 0 & -2 \\ -2 & 1 & 1 \end{bmatrix} \).

Step 1: Calculate the determinant.

From Example 2, \( \det(B) = 3 \). Since \( \det(B) \neq 0 \), the inverse exists.

Step 2: Calculate all cofactors \(C_{ij}\). We already calculated the first row cofactors in Example 2: \( C_{11} = 2, C_{12} = 1, C_{13} = 3 \). Now we need to calculate the rest:

  • \( C_{21} = (-1)^{2+1} \det \begin{bmatrix} -1 & 0 \\ 1 & 1 \end{bmatrix} = - ((-1 \times 1) - (0 \times 1)) = - (-1) = 1 \)
  • \( C_{22} = (-1)^{2+2} \det \begin{bmatrix} 2 & 0 \\ -2 & 1 \end{bmatrix} = (2 \times 1) - (0 \times (-2)) = 2 \)
  • \( C_{23} = (-1)^{2+3} \det \begin{bmatrix} 2 & -1 \\ -2 & 1 \end{bmatrix} = - ((2 \times 1) - ((-1) \times (-2))) = - (2 - 2) = 0 \)
  • \( C_{31} = (-1)^{3+1} \det \begin{bmatrix} -1 & 0 \\ 0 & -2 \end{bmatrix} = ((-1) \times (-2)) - (0 \times 0) = 2 \)
  • \( C_{32} = (-1)^{3+2} \det \begin{bmatrix} 2 & 0 \\ 3 & -2 \end{bmatrix} = - ((2 \times (-2)) - (0 \times 3)) = - (-4) = 4 \)
  • \( C_{33} = (-1)^{3+3} \det \begin{bmatrix} 2 & -1 \\ 3 & 0 \end{bmatrix} = (2 \times 0) - ((-1) \times 3) = 0 - (-3) = 3 \)
So, the matrix of cofactors is \( C = \begin{bmatrix} 2 & 1 & 3 \\ 1 & 2 & 0 \\ 2 & 4 & 3 \end{bmatrix} \).

Step 3: Find the adjugate \( \text{adj}(B) = C^T \).

\( \text{adj}(B) = C^T = \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 4 \\ 3 & 0 & 3 \end{bmatrix} \)

Step 4: Calculate \( B^{-1} = \frac{1}{\det(B)} \text{adj}(B) \).

\( B^{-1} = \frac{1}{3} \begin{bmatrix} 2 & 1 & 2 \\ 1 & 2 & 4 \\ 3 & 0 & 3 \end{bmatrix} = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ \frac{3}{3} & \frac{0}{3} & \frac{3}{3} \end{bmatrix} = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ 1 & 0 & 1 \end{bmatrix} \)

Thus, \( B^{-1} = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} & \frac{2}{3} \\ \frac{1}{3} & \frac{2}{3} & \frac{4}{3} \\ 1 & 0 & 1 \end{bmatrix} \).

Example 4: Verification (Optional, but Recommended)

You can verify by multiplying \( B \times B^{-1} \) and \( B^{-1} \times B \) to check if you get the \(3 \times 3\) Identity Matrix \( I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \). This is a good way to check for errors in your calculations, especially when learning. (Verification is left as an exercise).

4) Practice Questions 🎯

5.1 Fundamental – Build Skills

1. Calculate the determinant of matrix \(A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 2 & -1 \\ -1 & 1 & 0 \end{bmatrix}\).

2. Calculate the determinant of matrix \(B = \begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & -1 \\ 2 & -3 & 5 \end{bmatrix}\).

3. Calculate the determinant of matrix \(C = \begin{bmatrix} 3 & 2 & 1 \\ -1 & 0 & 2 \\ 4 & -2 & 3 \end{bmatrix}\).

4. Is matrix \(A = \begin{bmatrix} 2 & 0 & 1 \\ 1 & 2 & -1 \\ -1 & 1 & 0 \end{bmatrix}\) invertible? (Use your result from question 1).

5. Find the cofactor \(C_{12}\) for matrix \(B = \begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & -1 \\ 2 & -3 & 5 \end{bmatrix}\).

6. Find the cofactor \(C_{23}\) for matrix \(C = \begin{bmatrix} 3 & 2 & 1 \\ -1 & 0 & 2 \\ 4 & -2 & 3 \end{bmatrix}\).

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

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

9. Verify that your answer for the inverse of matrix \(D\) (from question 7) is correct by multiplying \(D\) and \(D^{-1}\) (at least check \(D \times D^{-1} = I_3\)).

10. Calculate the determinant of the Identity Matrix \(I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\). What do you notice?

5.2 Challenging – Push Limits 💪🚀

1. Calculate the determinant of \(A = \begin{bmatrix} x & y & 1 \\ 1 & 2 & 1 \\ 2 & 3 & 1 \end{bmatrix}\). For what values of \(x\) and \(y\) is \( \det(A) = 0 \)? What does this mean about the rows/columns of \(A\) in such cases?

2. For matrices \(B = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1 \end{bmatrix}\) and \(C = \begin{bmatrix} 1 & 1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{bmatrix}\) (diagonal and upper triangular matrices), calculate \( \det(B) \) and \( \det(C) \). Can you see a pattern for determinants of diagonal and triangular matrices?

3. Let \(A\) be a \(3 \times 3\) matrix with \( \det(A) = 4 \). Find \( \det(2A) \). (Hint: How does scalar multiplication affect the determinant of a \(3 \times 3\) matrix?).

4. (Conceptual) Explain why if a row (or column) of a matrix is all zeros, the determinant of the matrix is zero. (Think about cofactor expansion).

5. (Application - System of Equations) Consider the system of linear equations: \( 2x + y + z = 5 \) \( x - y + z = 1 \) \( 3x + y - z = 4 \) Represent this system in matrix form \(AX = B\), and use the inverse of matrix \(A\) (if it exists) to solve for \(X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\). (This might be computationally intensive by hand, but good practice! In Part 5, we may explore more efficient methods for larger systems).

6) Summary 🎉

  • Determinant of a 3x3 Matrix: Calculated using cofactor expansion along any row or column: \( \det(A) = \sum_{j=1}^{3} a_{ij} C_{ij} \) (row \(i\)) or \( \det(A) = \sum_{i=1}^{3} a_{ij} C_{ij} \) (column \(j\)).
  • Cofactor \(C_{ij}\): \( C_{ij} = (-1)^{i+j} M_{ij} \), where \(M_{ij}\) is the determinant of the \(2 \times 2\) minor matrix.
  • Invertibility: A \(3 \times 3\) matrix \(A\) is invertible if \( \det(A) \neq 0 \).
  • Adjugate of \(A\): \( \text{adj}(A) = C^T \), transpose of the cofactor matrix.
  • Inverse Formula: If \( \det(A) \neq 0 \), then \( A^{-1} = \frac{1}{\det(A)} \text{adj}(A) \).
  • Verification: Check if \( A \times A^{-1} = A^{-1} \times A = I_3 \) to verify the inverse.

Excellent work mastering determinants and inverses of \(3 \times 3\) matrices! You've significantly expanded your matrix algebra skills. These concepts are essential for various advanced mathematical and applied areas. Keep practicing to build speed and accuracy in calculations. Matrix Algebra will continue to unfold in future topics! 🌟 Get ready for more matrix adventures!

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