Question

Write the matrix in reduced row-echelon form. $$ \left[\begin{array}{rrr} 4 & 4 & 8 \\ 1 & 2 & 2 \\ -3 & 6 & -9 \end{array}\right] $$

Short answer

The matrix in reduced row-echelon form is: \[\begin{array}{ccc} 1 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\]

Step-by-step solution

Identify the first (pivot) element

First, a pivot point needs to be identified. That's the \(4\) in the top left of the matrix. Note that the pivot row is the first row and pivot column is the first column.

Turn the pivot to 1

Next, the aim is to make the pivot to \(1\). This can be achieved by dividing the first row by \(4\), so the first row becomes \([1, 1, 2]\). This step changes the matrix into \[\begin{array}{rrr} 1 & 1 & 2 \ 1 & 2 & 2 \ -3 & 6 & -9 \end{array}\]

Make the elements below the pivot 0

Then, subtract the first row from the second row and add three times the first row to the third row. This will make the first elements of the second and third rows \(0\). The matrix now becomes:\[\begin{array}{ccc} 1 & 1 & 2 \ 0 & 1 & 0 \ 0 & 9 & 3 \end{array}\]

Repeat for the next submatrix

The aim is to apply similar steps to the next submatrix (i.e., bottom right \(2x2\) matrix). This pivot point is the \(1\) in the middle of the matrix. The goal is to make the third element of the second row \(0\). Subtract nine times the second row from the third row, and the matrix now becomes:\[\begin{array}{ccc} 1 & 1 & 2 \ 0 & 1 & 0 \ 0 & 0 & -3 \end{array}\]

Scale the last row

Finally, divide the last row by \(-3\) to get another simplified row with the first non-zero element equals to \(1\) and remaining elements equal to \(0\). This results in:\[\begin{array}{ccc} 1 & 1 & 2 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\]

Apply Back Substitution

This involves making all entries above the pivot element to be \(0\). Begin by subtracting the third row from the first row. The matrix will be: \[\begin{array}{ccc} 1 & 1 & 1\ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\] then subtract the second row from the first row. The matrix now becomes:\[\begin{array}{ccc} 1 & 0 & 1 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\] This is the desired matrix in reduced row-echelon form.

Key concepts

Matrix Transformation

Matrix transformation involves changing the structure of a matrix using various strategies to achieve a particular form, like the reduced row-echelon form. When you work on matrices, think of them as grids of numbers. Transforming them can make solving linear equations much more manageable. The key in matrix transformation is applying operations systematically to arrive at the desired matrix structure. Reducing a matrix involves various techniques like swapping rows, scaling rows, or adding and subtracting rows. Each of these methods has specific rules to ensure the integrity and logical consistency of the matrix. Steps in matrix transformation are aimed at simplifying the matrix to fetch easily interpretable solutions from systems of linear equations.

Pivot Element

The pivot element is a crucial component, acting as a reference point when transforming a matrix. You can think of it as the leading hero playing a significant role in each row. Typically, the pivot element is the first non-zero number in a row, and acts as a guide to perform other actions like row operations.

• Start by identifying the first non-zero element on the left in the row to act as the pivot.
• Ensure that each pivot is equal to 1 by scaling the row, if necessary.
• Use the pivot to perform operations that will create zeros below it, establishing a stair-step pattern moving from left to right in the matrix.

This systematic selection and standardization process ensures the matrix moves toward the reduced row-echelon form, providing clarity and ease in further mathematical problem-solving.

Back Substitution

Back substitution is a technique used to solve equations once a matrix is in reduced row-echelon form. It's like a reverse process where you start solving from the bottom upwards, which simplifies finding solutions to the equations.

• You begin with the bottom-most pivot row, which in a solved matrix, will often provide the most direct answer for one of the variables.
• By helping to establish relationships, each pivot element assists in eliminating variables from rows above it, ultimately solving the system.
• Slowly move up each row by substituting the found values back into the equations to uncover the remaining unknowns.

Back substitution complements the matrix transformation process by delivering clear, concise, and neat solutions to an otherwise complex tangled set of equations.

Row Operations

Row operations are fundamental steps used to manipulate matrices into reduced row-echelon form. Think of them as strategic moves on a chessboard, maneuvering pieces to win the game of linear equations. There are three primary types of row operations:

• Swapping Rows: Switch two rows to move a better pivot into position, sometimes simplifying the process.
• Scaling Rows: Multiply all elements of a row by a non-zero scalar to adjust pivot elements to optimize further operations.
• Row Addition or Subtraction: Add or subtract a multiple of one row from another to create zeros above and below the pivot elements.

Mastery of these operations allows for seamless navigation through matrix transformations, efficiently guiding matrices to the much more accessible reduced row-echelon form.