Problem 53 - Algebra and Trigonometry

Question

Explaining the Concepts In your own words, describe each of the three matrix row operations. Give an example with each of the operations.

Step-by-Step Solution

Answer

Row operations in a matrix include: row exchange where the positions of rows are swapped, row scaling where a row is multiplied by a scalar value, and row addition where one row is added to another.

Step 1: Operation 1: Row Exchange

This operation involves swapping of rows position. This helps to simplify calculations when solving systems of linear equations. Let's consider a 3x3 matrix: \[ \begin{matrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{matrix} \]. To switch the first and third rows, you would get the following matrix: \[ \begin{matrix} 7 & 8 & 9 \ 4 & 5 & 6 \ 1 & 2 & 3 \end{matrix} \]. In this operation, no change is observed in the determinant value of the matrix.

Step 2: Operation 2: Row Scaling

Row scaling involves multiplying a row by a non-zero scalar. It's used to scale equations in a system to make the problem easier to handle. For example, consider the matrix \[ \begin{matrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{matrix} \]. Multiplying the first row by 2 yields the following matrix: \[ \begin{matrix} 2 & 4 & 6 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{matrix} \]. Keep in mind that this operation affects the determinant of the matrix as it gets scaled by the same factor.

Step 3: Operation 3: Row Addition

Row Addition involves adding a row to another row. This operation is mainly used to eliminate variables when solving a system of linear equations. Let's consider a 3x3 matrix: \[ \begin{matrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{matrix} \]. If the first row is added to the second row, then we get the following matrix: \[ \begin{matrix} 1 & 2 & 3 \ 5 & 7 & 9 \ 7 & 8 & 9 \end{matrix} \]. This operation does not change the determinant of the matrix.

Key Concepts

Matrix Row Exchange

Think about matrix row exchange as the act of swapping places of rows in a matrix, much like exchanging two cards in your hand. This operation is quite useful when dealing with systems of linear equations. It helps rearrange the position of equations to make problem-solving more straightforward.

For example, consider a matrix like this:

• Row 1: [1, 2, 3]
• Row 2: [4, 5, 6]
• Row 3: [7, 8, 9]

To perform a row exchange, swap Row 1 and Row 3, making the matrix look like this:

• Row 1: [7, 8, 9]
• Row 2: [4, 5, 6]
• Row 3: [1, 2, 3]

Notice the numbers have changed places, but the matrix's determinant remains the same. This means the swap doesn't affect the fundamental properties of the matrix, making row exchange a safe and useful technique in matrix manipulations.

Matrix Row Scaling

Matrix row scaling involves multiplying an entire row by a number, known as a scalar. Imagine you're pulling on a stretching rubber band— the same is done to the row's values. Each number in the row is increased or decreased in size while maintaining their relationships to one another.

Let's say you have the matrix:

• Row 1: [1, 2, 3]
• Row 2: [4, 5, 6]
• Row 3: [7, 8, 9]

Imagine multiplying the first row by 2. The results turn into:

• Row 1: [2, 4, 6]
• Row 2: [4, 5, 6]
• Row 3: [7, 8, 9]

Notice how each number in Row 1 doubles. While this aids in simplifying calculations, it's essential to know that this operation changes the matrix's determinant by the same multiplicative factor, which is crucial in solving or analyzing the system.

Matrix Row Addition

Matrix row addition is like mixing ingredients to create a new recipe. Here, you are adding one row to another, adjusting the matrix's numbers and assisting in eliminating variables systematically. Picture it as blending one row with another to achieve a specific goal in your computations.

Consider this simple matrix setup:

• Row 1: [1, 2, 3]
• Row 2: [4, 5, 6]
• Row 3: [7, 8, 9]

Adding Row 1 to Row 2 gives you:

• Row 1: [1, 2, 3]
• Row 2: [5, 7, 9]
• Row 3: [7, 8, 9]

This strategic combination helps maintain the matrix's structure and solve variables one by one without influencing the determinant, keeping the matrix stable. This operation is vital when performing techniques like Gaussian elimination to simplify and solve systems.