Test yourself: Gauss-Jordan method for 3x3 inverse matrix problems

📚 Quick Study Guide

• 🔢 Matrix Augmentation: Start by augmenting the given 3x3 matrix $A$ with the 3x3 identity matrix $I$, forming $[A | I]$.
• ➗ Row Operations: Apply elementary row operations to transform the matrix $A$ into the identity matrix. The same operations must be applied to the identity matrix on the right. These operations include:
  • 🔄 Swapping two rows.
  • ✖️ Multiplying a row by a non-zero scalar.
  • ➕ Adding a multiple of one row to another.
• 🎯 Target Form: Aim to get the left side to be the identity matrix. After the row operations, the augmented matrix will look like $[I | A^{-1}]$, where $A^{-1}$ is the inverse of $A$.
• ✅ Inverse Matrix: The matrix on the right side after the transformation is the inverse of the original matrix $A$.
• 📝 Non-Invertible Matrices: If, during the process, you obtain a row of zeros on the left side, the matrix is singular and does not have an inverse.
• 💡 Formula Recap: The Gauss-Jordan method essentially solves $AX = I$ for $X$, where $X = A^{-1}$.

Practice Quiz

1. Which of the following 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 eigenvalues of the matrix.
   4. D. Transpose the matrix.
2. What type of row operation is allowed when using the Gauss-Jordan method?
   1. A. Multiplying two rows together.
   2. B. Adding a multiple of one row to another.
   3. C. Taking the square root of all elements in a row.
   4. D. Cubing all elements in a row.
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 inverse matrix is the identity matrix.
   2. B. The matrix is invertible.
   3. C. The matrix is singular and does not have an inverse.
   4. D. You need to start over with different row operations.
4. What is the final form of the augmented matrix after successfully applying the Gauss-Jordan method to find the inverse?
   1. A. $[A | I]$
   2. B. $[I | A]$
   3. C. $[I | A^{-1}]$
   4. D. $[A^{-1} | I]$
5. Which row operation is NOT typically used in the Gauss-Jordan method?
   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. Taking the logarithm of all elements in a row.
6. Suppose you have the matrix $\begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 1 \end{bmatrix}$. After applying Gauss-Jordan, what should the left side of the augmented matrix look like?
   1. A. $\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}$
   2. B. $\begin{bmatrix} 1 & 2 & 3 \ 0 & 1 & 4 \ 0 & 0 & 1 \end{bmatrix}$
   3. C. $\begin{bmatrix} 0 & 0 & 1 \ 0 & 1 & 0 \ 1 & 0 & 0 \end{bmatrix}$
   4. D. $\begin{bmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{bmatrix}$
7. In the Gauss-Jordan method, what are you essentially solving for when finding the inverse of a matrix $A$?
   1. A. $AX = 0$
   2. B. $AX = I$
   3. C. $A^{-1}X = I$
   4. D. $A + X = I$