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Chapter 2: Problem 14

Find a sequence of elementary matrices that can be used to write the matrix in row-echelon form. $$ \left[\begin{array}{rrrr} 0 & 3 & -3 & 6 \\ 1 & -1 & 2 & -2 \\ 0 & 0 & 2 & 2 \end{array}\right] $$

Short Answer

Expert verified
The sequence of elementary matrices applied are: E1 = \[ \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} \] E2 = \[ \begin{pmatrix} 1 & 0 & 0 \ 0 & \frac{1}{3} & 0 \ 0 & 0 & 1 \end{pmatrix} \] E3 = \[ \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & \frac{1}{2} \end{pmatrix} \]

Step by step solution

01

Apply Elementary Row Operation to Swap Rows

We want to start with a non-zero element in the top left corner of the matrix, so swap first and second rows. The matrix becomes: \[ \begin{pmatrix} 1 & -1 & 2 & -2 \ 0 & 3 & -3 & 6 \ 0 & 0 & 2 & 2 \end{pmatrix} \] The corresponding elementary matrix is: \[ E_1 = \begin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{pmatrix} \]
02

Apply Elementary Row Operation to Make Leading Coefficient 1

The pivot, or the leading coefficient of the second row, should be 1. So, divide all elements in the second row by 3. The matrix becomes: \[ \begin{pmatrix} 1 & -1 & 2 & -2 \ 0 & 1 & -1 & 2 \ 0 & 0 & 2 & 2 \end{pmatrix} \] The corresponding elementary matrix is: \[ E_2 = \begin{pmatrix} 1 & 0 & 0 \ 0 & \frac{1}{3} & 0 \ 0 & 0 & 1 \end{pmatrix} \]
03

Apply Elementary Row Operation to Make Zeroes Below the Pivot

To obtain zeros below the pivot of the last row, divide all the elements in the third row by 2. The matrix becomes: \[ \begin{pmatrix} 1 & -1 & 2 & -2 \ 0 & 1 & -1 & 2 \ 0 & 0 & 1 & 1 \end{pmatrix} \] The corresponding elementary matrix is: \[ E_3 = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & \frac{1}{2} \end{pmatrix} \]

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Key Concepts

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

Row-Echelon Form
The row-echelon form of a matrix is a way to simplify a matrix so it's easier to solve. In this form, each row of the matrix has more leading zeros than the row above it, which creates a "stair-step" pattern. This systematic arrangement allows us to easily identify pivot positions and solve the system of equations represented by the matrix.
  • The first non-zero number in each row, which is called a pivot, is always to the right of the pivot in the row above it.
  • All the elements below each pivot in a column are zeros.
  • Any rows consisting entirely of zeros are at the bottom of the matrix.

Converting a matrix to row-echelon form involves a series of strategic row operations to position pivots correctly. The ultimate goal is to simplify solving systems of linear equations by transforming the matrix into a form that is much easier to work with.
Elementary Row Operations
Elementary row operations are the tools we use to transform matrices into row-echelon form. These operations can be applied without changing the solution set of the corresponding system of equations. Here are the types of operations you can perform:
  • Swapping two rows.
  • Multiplying a row by a non-zero constant.
  • Adding or subtracting a multiple of one row to another row.

For example, in the solved exercise, swapping and scaling operations were performed to achieve the desired echelon form. Initially, rows were swapped to ensure a non-zero element (a pivot) in the top left corner. Then, rows were scaled (divided by a constant) to simplify the pivots to 1, making further calculations more straightforward. These systematic operations allow us to step closer to the row-echelon form.
Pivot Positions
Pivot positions are critical in understanding matrix operations, especially when converting a matrix to its row-echelon form. A pivot is the first non-zero element in a row, and it plays a crucial role in determining the structure of the matrix:
  • They dictate the row operations needed to achieve the row-echelon form.
  • The location of a pivot affects how the rows will be manipulated (swapped, scaled, etc.).
  • They are essential for determining the number of free variables in the system of equations.

In the context of solving matrices, to ensure that the matrix is in row-echelon form, identifying and working with the correct pivot positions is key. These positions guide readers on which columns should contain leading ones after the matrix transformation. This makes it easier to interpret the matrix and solve it methodically.

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Most popular questions from this chapter

CAPSTONE (a) Explain how to find an elementary matrix. (b) Explain how to use elementary matrices to find an LU-factorization of a matrix. (c) Explain how to use \(L U\) -factorization to solve a linear system.

Find the least squares regression line. $$ (-2,0),(-1,1),(0,1),(1,2) $$

Wildlife \(A\) wildlife management team studied the reproduction rates of deer in three tracts of a wildlife preserve. The team recorded the number of females \(x\) in each tract and the percent of females \(y\) in each tract that had offspring the following year. The table shows the results. $$ \begin{array}{l|ccc} \hline \text {Number, }, x & 100 & 120 & 140 \\ \text {Percent, } y & 75 & 68 & 55 \\ \hline \end{array} $$ (a) Find the least squares regression line for the data. (b) Use a graphing utility to graph the model and the data in the same viewing window. (c) Use the model to create a table of estimated values for y. Compare the estimated values with the actual data. (d) Use the model to estimate the percent of females that had offspring when there were 170 females. (e) Use the model to estimate the number of females when \(40 \%\) of the females had offspring.

Prove that if \(A\) and \(B\) are \(n \times n\) skew-symmetric matrices, then \(A+B\) is skew-symmetric.

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) Matrix multiplication is commutative. (b) If the matrices \(A, B,\) and \(C\) satisfy \(A B=A C,\) then \(B=C\) (c) The transpose of the sum of two matrices equals the sum of their transposes.
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