Question

If row 2 of a 3 by 3 determinant is multiplied by \(k, k \neq 0,\) and the result is added to the entries in row 1 , prove that there is no change in the value of the determinant.

Short answer

The value of the determinant remains unchanged.

Step-by-step solution

- Write the determinant

Consider a 3x3 determinant \[ D = \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} \]

- Apply row operation

Perform the row operation given in the problem: Multiply row 2 by k and add the result to row 1. This gives us a new determinant: \[ D' = \begin{vmatrix} a + kd & b + ke & c + kf \ d & e & f \ g & h & i \end{vmatrix} \]

- Use properties of determinants

By the properties of determinants, adding a multiple of one row to another row does not change the value of the determinant. Hence, \[ \begin{vmatrix} a + kd & b + ke & c + kf \ d & e & f \ g & h & i \end{vmatrix} = \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} \]

- Conclude the proof

Since the modified determinant value is equal to the original determinant value, we have shown that the determinant remains unchanged after the described row operation.

Key concepts

3x3 Matrix

A 3x3 matrix is a square matrix that contains 3 rows and 3 columns. Each element in the matrix can be represented depending on its row and column index. For example, the element in the first row and first column is often denoted as \(a_{11}\). For our exercise, the matrix is written as follows:
\[ \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} \]

This structure allows us to perform various operations such as addition, subtraction, multiplication, and more. Understanding the positioning of each element is critical as it plays a key role in row operations and calculating the determinant.

Row Operations

Row operations are techniques used to simplify calculations or solve systems of linear equations. Typical row operations include:

• Swapping two rows
• Multiplying a row by a nonzero scalar
• Adding or subtracting a multiple of one row to another row

In our exercise, we specifically use the operation of multiplying the second row by a scalar \(k\) and then adding this result to the first row. Our original matrix is:
\[ \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} \]
After performing the operation, the new matrix becomes:
\[ \begin{vmatrix} a + kd & b + ke & c + kf \ d & e & f \ g & h & i \end{vmatrix} \]%
Understanding these operations is crucial for manipulating matrices and calculating their determinants.

Properties of Determinants

Determinants have several key properties that are helpful when simplifying and solving problems. Some important properties include:

• The determinant of a square matrix changes sign if any two rows (or columns) are swapped.
• The determinant is zero if any row (or column) is a linear combination of the others.
• Adding a multiple of one row to another does not change the determinant.

In the given exercise, we used the third property. By multiplying row 2 by \(k\) and adding it to row 1, the determinant remains unchanged:
\[ \begin{vmatrix} a + kd & b + ke & c + kf \ d & e & f \ g & h & i \end{vmatrix} = \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} \]%
This property is immensely useful in various proofs and calculations involving determinants.

Proof Techniques

Proof techniques in linear algebra often involve carefully applying operations and properties to demonstrate a particular result. In our exercise, we wanted to show that a specific row operation does not alter the determinant. Our proof involved the following steps:
1. Writing the original determinant.
2. Performing the row operation of multiplying row 2 by \(k\) and adding it to row 1.
3. Applying a known property of determinants that confirms the operation does not change the determinant's value.
These steps showed that:\[ \begin{vmatrix} a + kd & b + ke & c + kf \ d & e & f \ g & h & i \end{vmatrix} = \begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} \]%
Mastering these techniques is essential for dealing with more complex matrix and determinant problems. They provide a structured way to tackle proofs methodically and help validate our results.