# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: ## **Echelon Form:** The **echelon form** of a matrix is a matrix that has the following properties: 1. All rows containing non-zero entries are above rows containing only zero entries. 2. The leading entry of each non-zero row (i.e., the first non-zero number from the left) is 1 and appears to the right of the leading entry of the previous row. 3. The entries below and above each leading 1 are zero. Example of transforming a matrix to echelon form: $$ \text{Starting matrix:} \quad \left[\begin{array}{ccc} 1 & 3 & 5 \\ 2 & 6 & 8 \\ 4 & 9 & 7 \end{array}\right] $$ Performing row operations: - Subtract 2 times Row 1 from Row 2. - Subtract 4 times Row 1 from Row 3. **Resulting echelon form:** $$ \left[\begin{array}{ccc} 1 & 3 & 5 \\ 0 & 0 & -2 \\ 0 & 0 & -3 \end{array}\right] $$ ## **Reduced Echelon Form:** The **reduced echelon form** is a stricter version of echelon form. In addition to the rules for echelon form: 1. The leading 1 in each row is the only non-zero entry in its column. Example of transforming a matrix to reduced echelon form: $$ \text{Starting matrix:} \quad \left[\begin{array}{ccc} 1 & 3 & 5 \\ 0 & 2 & 4 \\ 0 & 0 & 1 \end{array}\right] $$ To reduce: - Divide Row 2 by 2 to make the leading entry 1. - Subtract appropriate multiples of Row 2 from Row 1. Resulting reduced echelon form: $$ \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ ## **Row Operations:** Row operations are used to manipulate the rows of a matrix and are crucial in solving systems of linear equations, finding the rank, and transforming matrices to echelon and reduced echelon forms. The three types of row operations are: 1. **Row switching**: Swap two rows. 2. **Row scaling**: Multiply a row by a non-zero scalar. 3. **Row addition**: Add a multiple of one row to another. Example of a row operation: $$ \text{Row 2 becomes Row 2 minus 3 times Row 1:} \quad \left[\begin{array}{ccc} 1 & 3 & 5 \\ 0 & -1 & -7 \\ 4 & 9 & 7 \end{array}\right] $$ ## **Rank of a Matrix:** The **rank** of a matrix is the number of non-zero rows in its echelon form. It represents the number of linearly independent rows or columns in a matrix. Example: For the matrix: $$ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 0 & 0 \end{array}\right] $$ Performing row operations: - Row 2 becomes Row 2 minus 2 times Row 1. The resulting echelon form: $$ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] $$ The rank is 1, because there is only one non-zero row. --- # **Important Formulas** Below are the key formulas used in this exercise: - **Echelon Form**: A matrix is in echelon form if: 1. All non-zero rows are above any zero rows. 2. The leading entry (the first non-zero number from the left) in each row is 1. 3. The leading entry in each row appears to the right of the leading entry in the previous row. 4. All entries in the column below and above a leading 1 are zero. Example: $$ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{array}\right] $$ - **Reduced Echelon Form**: A matrix is in reduced echelon form if: 1. It is in echelon form. 2. The leading entry in each row is the only non-zero entry in its column. Example: $$ \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] $$ - **Inverse of a Matrix**: The inverse of a matrix $A$ is denoted $A^{-1}$, and the matrix satisfies the condition: $$ A \times A^{-1} = A^{-1} \times A = I $$ Where $I$ is the identity matrix. Example: For $A = \left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right]$, the inverse is: $$ A^{-1} = \frac{1}{(1 \cdot 4 - 2 \cdot 3)} \left[\begin{array}{cc} 4 & -2 \\ -3 & 1 \end{array}\right] $$ Resulting in: $$ A^{-1} = \frac{1}{-2} \left[\begin{array}{cc} 4 & -2 \\ -3 & 1 \end{array}\right] $$ - **Rank of a Matrix**: The rank of a matrix is the number of non-zero rows in its echelon form. The rank represents the number of linearly independent rows or columns. Example: For the matrix: $$ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 0 & 0 & 0 \end{array}\right] $$ After performing row operations, the echelon form is: $$ \left[\begin{array}{ccc} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] $$ The rank is 1, since there is only one non-zero row. --- # **Summary** This exercise requires you to reduce matrices to their echelon and reduced echelon forms using row operations. You'll also compute the rank of matrices and find their inverses. Key techniques include: - Applying Gaussian elimination and row operations to reach echelon form. - Normalizing rows to achieve reduced echelon form. - Verifying matrix inverses by checking $A \times A^{-1} = I$. The exercise tests your understanding of matrix operations and their application in solving systems of linear equations. --- # **Reference Videos** ![](https://www.youtube.com/watch?v=XoCwteySKKg&pp=ygUTZXggMi41IG5iZiBjbGFzcyAxMQ%3D%3D)