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Chapter 6: Problem 21

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

Short Answer

Expert verified
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 by step solution

01

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: \[\left[\begin{array}{rrr|r}3 & -12 & 6 & 9 \ 1 & -4 & 4 & 0 \ 2 & 0 & 7 & 4\end{array}\right]\]
02

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: \[\left[\begin{array}{rrr|r}1 & -4 & 2 & 3 \ 1 & -4 & 4 & 0 \ 2 & 0 & 7 & 4\end{array}\right]\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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:

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Most popular questions from this chapter

In Exercises \(27-36,\) find (if possible): \(\begin{array}{llll}\text { a. } A B & \text { and } & \text { b. } B A\end{array}\) $$ A=\left[\begin{array}{rrr} 1 & -1 & 4 \\ 4 & -1 & 3 \\ 2 & 0 & -2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 1 & 1 & 0 \\ 1 & 2 & 4 \\ 1 & -1 & 3 \end{array}\right] $$

Use Cramer's rule to solve each system. $$ \begin{aligned}4 x-5 y-6 z &=-1 \\\x-2 y-5 z &=-12 \\\2 x-y &=7\end{aligned} $$

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