Grade 11 Unit 4: 4.5 Inverse of a Square Matrix of Order 2 and 3 and Exercise4. 10 & 4.9| Saquama

Introduction

This tutorial provides a step-by-step guide on how to find the inverse of square matrices of order 2 and 3, based on the concepts discussed in the Grade 11 Maths curriculum. Understanding matrix inverses is crucial for solving systems of equations and various applications in mathematics and engineering.

Step 1: Understanding Matrix Inverses

• The inverse of a matrix ( A ) is denoted as ( A^{-1} ).
• A matrix has an inverse only if it is non-singular, meaning its determinant is not zero.
• The product of a matrix and its inverse results in the identity matrix: [ A \cdot A^{-1} = I ]

Step 2: Finding the Inverse of a 2x2 Matrix

1. Identify the matrix: Let ( A = \begin{pmatrix} a & b \ c & d \end{pmatrix} ).

2. Calculate the determinant: [ \text{det}(A) = ad - bc ] Ensure that ( \text{det}(A) \neq 0 ).

3. Apply the formula for the inverse: If the determinant is not zero, the inverse is given by: [ A^{-1} = \frac{1}{\text{det}(A)} \begin{pmatrix} d & -b \ -c & a \end{pmatrix} ]

4. Example: For ( A = \begin{pmatrix} 2 & 3 \ 1 & 4 \end{pmatrix} ):

   • Calculate the determinant: ( \text{det}(A) = (2)(4) - (3)(1) = 8 - 3 = 5 ).
   • Find the inverse: [ A^{-1} = \frac{1}{5} \begin{pmatrix} 4 & -3 \ -1 & 2 \end{pmatrix} = \begin{pmatrix} 0.8 & -0.6 \ -0.2 & 0.4 \end{pmatrix} ]

Step 3: Finding the Inverse of a 3x3 Matrix

1. Identify the matrix: Let ( A = \begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} ).

2. Calculate the determinant using the formula: [ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) ] Ensure that ( \text{det}(A) \neq 0 ).

3. Find the matrix of minors:

   • Calculate the determinant of each 2x2 submatrix obtained by removing one row and one column.

4. Calculate the matrix of cofactors:

   • Apply the checkerboard pattern of signs to the matrix of minors.

5. Find the adjugate:

   • Transpose the matrix of cofactors.

6. Apply the formula for the inverse: [ A^{-1} = \frac{1}{\text{det}(A)} \cdot \text{adj}(A) ]

7. Example: For ( A = \begin{pmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 5 & 6 & 0 \end{pmatrix} ):

   • Calculate the determinant: ( \text{det}(A) = 1(1\cdot0 - 4\cdot6) - 2(0\cdot0 - 4\cdot5) + 3(0\cdot6 - 1\cdot5) = -24 + 40 - 15 = 1 ).
   • Find the inverse using the steps outlined above.

Conclusion

Finding the inverse of square matrices is a fundamental skill in mathematics, especially in solving linear equations. This guide provides the necessary steps to calculate the inverse of both 2x2 and 3x3 matrices effectively. Practice with various matrices to strengthen your understanding, and consider exploring real-world applications such as computer graphics or systems modeling where matrix operations are vital.