**Problem 22:**\[ \left[\begin{array}{cc} 3 & 2 \\ 5 & 4 \end{array}\right], \left[\begin{array}{cc} 3 & 2 \\ 5 + 3k & 4 + 2k \end{array}\right] \]This image shows two 2x2 matrices separated by a comma. The second matrix includes variables, implying a potential problem involving algebraic expressions or transformations in linear algebra. The first matrix is purely numeric, while the second matrix has elements that are functions of \(k\):- The first matrix is: \[ \left[\begin{array}{cc} 3 & 2 \\ 5 & 4 \end{array}\right] \]- The second matrix is: \[ \left[\begin{array}{cc} 3 & 2 \\ 5 + 3k & 4 + 2k \end{array}\right] \]Detailed examination of the relationship between these two matrices might involve operations such as addition, multiplication, or finding values of \(k\) that satisfy a given condition. **Exploring the Effect of Elementary Row Operations on the Determinant of a Matrix**In Exercises 19-24, you are tasked with exploring the effect of an elementary row operation on the determinant of a matrix. For each case, state the row operation and describe how it affects the determinant.When performing these exercises, follow these steps:1. **Identify the Row Operation:** Clearly define which elementary row operation you are using in the given exercise. Elementary row operations include: - **Swapping Two Rows:** Exchanging two rows in a matrix. - **Multiplying a Row by a Scalar:** Multiplying all elements of a row by a non-zero scalar. - **Adding a Multiple of One Row to Another Row:** Adding a multiple of one row to another row.2. **Analyze the Determinant:** Determine how the specific row operation affects the determinant of the matrix. Remember these key rules for the determinant: - **Swapping Rows:** Swapping two rows of a matrix changes the sign of the determinant. - **Multiplying a Row by a Scalar:** Multiplying a row by a scalar (k) multiplies the determinant by that same scalar (k). - **Adding a Multiple of One Row to Another:** Adding a multiple of one row to another row does not change the determinant.3. **Provide a Detailed Description:** In your solution, clearly describe the row operation applied and explain how it influences the determinant of the matrix.By following these guidelines, you'll gain a deeper understanding of how elementary row operations impact the determinant, providing you with essential tools for matrix manipulation and linear algebra problem-solving.

Name: **Problem 22:**\[ \left[\begin{array}{cc} 3 & 2 \\ 5 & 4 \end{array}\
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: **Problem 22:**\[ \left[\begin{array}{cc} 3 & 2 \\ 5 & 4 \end{array}\right], \left[\begin{array}{cc} 3 & 2 \\ 5 + 3k & 4 + 2k \end{array}\right] \]This image shows two 2x2 matrices separated by a comma. The second matrix includes variables, implying