Problem 21 - College Algebra

Question

In Exercises \(19-24\), perform each matrix row operation and write the new matrix. \(\left[\begin{array}{rrr|r}3 & -12 & 6 & 9 \\ 1 & -4 & 4 & 0 \\ 2 & 0 & 7 & 4\end{array}\right] \quad \frac{1}{3} R_{1}\)

Step-by-Step Solution

Answer

The new matrix after performing the operation \(\frac{1}{3} R_{1}\) is \[\left[\begin{array}{rrr|r}1 & -4 & 2 & 3 \ 1 & -4 & 4 & 0 \ 2 & 0 & 7 & 4\end{array}\right]\].

Step 1: Identify the operation

The operation to be performed here is \(\frac{1}{3} R_{1}\). This means that each element in row 1 is to be divided by 3. The matrix provided is: \[\left[\begin{array}{rrr|r}3 & -12 & 6 & 9 \ 1 & -4 & 4 & 0 \ 2 & 0 & 7 & 4\end{array}\right]\]

Step 2: Perform the operation

Perform the operation \(\frac{1}{3} R_{1}\) on the matrix, i.e., divide each element in the first row by 3. This gives us a new matrix as: \[\left[\begin{array}{rrr|r}1 & -4 & 2 & 3 \ 1 & -4 & 4 & 0 \ 2 & 0 & 7 & 4\end{array}\right]\]

Key Concepts

Elementary Row Operations

In linear algebra, mastering elementary row operations is a critical skill that serves as the foundation for many complex procedures such as solving systems of equations, finding determinants, and determining the rank of a matrix. Elementary row operations consist of three types: