Solved examples: Inverse of a 3x3 matrix with Gauss-Jordan method

📚 Quick Study Guide

• 🔢 The inverse of a matrix $A$, denoted as $A^{-1}$, is a matrix such that $A \cdot A^{-1} = I$, where $I$ is the identity matrix.
• ➗ The Gauss-Jordan method involves augmenting the given matrix $A$ with the identity matrix $I$ to form $[A|I]$.
• 🔨 Apply elementary row operations to transform $A$ into $I$. The matrix on the right side will then be $A^{-1}$, resulting in $[I|A^{-1}]$.
• ➕ Elementary row operations include: swapping rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another.
• 🧭 If at any point during the row operations, a row of zeros appears on the left side (the original matrix side), the matrix $A$ is singular and does not have an inverse.

Practice Quiz

1. Question 1: What is the first step in finding the inverse of a 3x3 matrix using the Gauss-Jordan method?
   1. A. Calculate the determinant of the matrix.
   2. B. Augment the matrix with the identity matrix.
   3. C. Find the transpose of the matrix.
   4. D. Calculate the cofactor matrix.
2. Question 2: Which of the following is NOT an elementary row operation?
   1. A. Swapping two rows.
   2. B. Multiplying a row by a non-zero scalar.
   3. C. Adding a multiple of one row to another.
   4. D. Calculating the adjoint of the matrix.
3. Question 3: If, during the Gauss-Jordan method, you obtain a row of zeros on the left side of the augmented matrix, what does this indicate?
   1. A. The matrix has an inverse and you should continue with row operations.
   2. B. The matrix is singular and does not have an inverse.
   3. C. You made a mistake in your calculations.
   4. D. The determinant of the matrix is 1.
4. Question 4: What is the identity matrix $I$ used for in the Gauss-Jordan method when finding the inverse of a matrix?
   1. A. It is used to calculate the determinant.
   2. B. It is used to augment the original matrix.
   3. C. It is used to find the transpose.
   4. D. It is used to solve for eigenvalues.
5. Question 5: After performing row operations, if the left side of the augmented matrix becomes the identity matrix, what does the right side represent?
   1. A. The original matrix.
   2. B. The transpose of the original matrix.
   3. C. The inverse of the original matrix.
   4. D. The determinant of the original matrix.
6. Question 6: Which row operation will transform the following matrix \[\begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}\] into a matrix with a '1' in the top left corner?
   1. A. $R_1 \rightarrow R_1 - 2R_2$
   2. B. $R_1 \rightarrow \frac{1}{2}R_1$
   3. C. $R_2 \rightarrow R_2 - \frac{1}{2}R_1$
   4. D. $R_1 \leftrightarrow R_2$
7. Question 7: What is the purpose of elementary row operations in the Gauss-Jordan method?
   1. A. To change the determinant of the matrix.
   2. B. To transform the original matrix into the identity matrix.
   3. C. To find the eigenvalues of the matrix.
   4. D. To calculate the trace of the matrix.