Math Problem Statement

We need to pivot on the entry in Row 2, Column 1. Remember that pivoting means: the pivot element becomes 1 AND all entries above/below the pivot element become 0. Let's first make the pivot element 1 by multiplying all entries in Row 2 by a non-zero constant. Fill in the blank with the appropriate constant.

3 1 2

2 4 8

Solution

To make the pivot element in Row 2, Column 1 equal to 1, we need to find the constant that will multiply all entries in Row 2.

The pivot element is currently 2, and we want to make it 1. So, we need to multiply Row 2 by the reciprocal of 2, which is 12\frac{1}{2}.

This will make the pivot element equal to 1.

Let's perform this operation:

Multiply Row 2 by 12\frac{1}{2}:

Row2=(2,4,8)×12=(1,2,4)Row 2 = \left( 2, 4, 8 \right) \times \frac{1}{2} = \left( 1, 2, 4 \right)

So, the constant to multiply Row 2 by is 12\frac{1}{2}.

The new matrix after the operation would look like:

3 & 1 & 2 \\ 1 & 2 & 4 \end{bmatrix}$$ Let me know if you'd like further explanation or have any questions! Here are some related questions: 1. What is the next step after pivoting in Gaussian elimination? 2. How do we handle the entries above the pivot in Row 1? 3. How can we perform pivoting in a 3x3 matrix? 4. What happens if we choose the wrong pivot element? 5. How do you determine if a system of equations has a unique solution after performing Gaussian elimination? **Tip:** Always check if the pivot element is non-zero before proceeding with the pivoting step, as dividing by zero is not valid.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Gaussian Elimination
Matrix Operations

Formulas

Pivoting step in Gaussian elimination: Row = Row × (1/pivot element)

Theorems

Gaussian Elimination Theorem

Suitable Grade Level

Grades 10-12