Question

Use Gaussian Elimination to put the given matrix into reduced row echelon form. $$\left[\begin{array}{cccc}4 & 1 & 8 & 15 \\ 1 & 1 & 2 & 7 \\ 3 & 1 & 5 & 11\end{array}\right]$$

Short answer

The matrix in reduced row echelon form is:\[ \begin{bmatrix} 1 & 0 & 0 & \frac{55}{12} \\ 0 & 1 & 0 & \frac{13}{3} \\ 0 & 0 & 1 & -1 \end{bmatrix} \]

Step-by-step solution

Identify the First Pivot

Begin with the first column. The largest absolute value in the first column is in the first row, so the pivot position is already set at (1,1) with a value of 4. No row swapping is necessary.

Make the First Pivot a 1

Transform the pivot at (1,1) from 4 to 1 by dividing the entire first row by 4:\[ R_1 = \frac{1}{4}R_1 \]After the operation, the matrix becomes:\[ \begin{bmatrix} 1 & \frac{1}{4} & 2 & \frac{15}{4} \ 1 & 1 & 2 & 7 \ 3 & 1 & 5 & 11 \end{bmatrix} \]

Eliminate Entries Below the First Pivot

Subtract multiples of the first row from the other rows to create zeros in the first column below the first pivot:For the second row:\[ R_2 = R_2 - R_1 \] becomes:\[ \begin{bmatrix} 0 & \frac{3}{4} & 0 & \frac{13}{4} \end{bmatrix} \]For the third row:\[ R_3 = R_3 - 3R_1 \] becomes:\[ \begin{bmatrix} 0 & \frac{1}{4} & -1 & \frac{7}{4} \end{bmatrix} \]The matrix now is:\[ \begin{bmatrix} 1 & \frac{1}{4} & 2 & \frac{15}{4} \ 0 & \frac{3}{4} & 0 & \frac{13}{4} \ 0 & \frac{1}{4} & -1 & \frac{7}{4} \end{bmatrix} \]

Identify and Make the Second Pivot a 1

The pivot in the second row is at (2,2) with a value of \(\frac{3}{4}\). Make this pivot a 1 by dividing the entirety of the second row by \(\frac{3}{4}\):\[ R_2 = \frac{4}{3}R_2 \]After this operation, the matrix becomes:\[ \begin{bmatrix} 1 & \frac{1}{4} & 2 & \frac{15}{4} \ 0 & 1 & 0 & \frac{13}{3} \ 0 & \frac{1}{4} & -1 & \frac{7}{4} \end{bmatrix} \]

Eliminate Entries Below the Second Pivot

Subtract \(\frac{1}{4}\) times the second row from the third row to eliminate the entry below the pivot in the second row:\[ R_3 = R_3 - \frac{1}{4}R_2 \] which becomes:\[ \begin{bmatrix} 0 & 0 & -1 & \frac{12}{3} \end{bmatrix} \]The matrix now is:\[ \begin{bmatrix} 1 & \frac{1}{4} & 2 & \frac{15}{4} \ 0 & 1 & 0 & \frac{13}{3} \ 0 & 0 & -1 & 1 \end{bmatrix} \]

Make the Third Pivot a 1

The pivot in the third row is at (3,3) with a value of -1. Multiply the entire third row by -1 to turn this pivot into 1:\[ R_3 = -1R_3 \]After this operation, the matrix becomes:\[ \begin{bmatrix} 1 & \frac{1}{4} & 2 & \frac{15}{4} \ 0 & 1 & 0 & \frac{13}{3} \ 0 & 0 & 1 & -1 \end{bmatrix} \]

Eliminate Entries Above the Third Pivot

Use row operations to create zeros above the pivot in the third row:For the first row:\[ R_1 = R_1 - 2R_3 \] becomes:\[ \begin{bmatrix} 1 & \frac{1}{4} & 0 & \frac{19}{4} \end{bmatrix} \]The matrix now is:\[ \begin{bmatrix} 1 & \frac{1}{4} & 0 & \frac{19}{4} \ 0 & 1 & 0 & \frac{13}{3} \ 0 & 0 & 1 & -1 \end{bmatrix} \]

Eliminate Entries Above the Second Pivot

Use row operations to create zeros above the second pivot:For the first row:\[ R_1 = R_1 - \frac{1}{4}R_2 \] becomes:\[ \begin{bmatrix} 1 & 0 & 0 & \frac{55}{12} \end{bmatrix} \]The final matrix in reduced row echelon form is:\[ \begin{bmatrix} 1 & 0 & 0 & \frac{55}{12} \ 0 & 1 & 0 & \frac{13}{3} \ 0 & 0 & 1 & -1 \end{bmatrix} \]

Key concepts

Pivoting

When dealing with systems of linear equations, pivoting is a crucial step in the Gaussian Elimination process. **But what is pivoting?** In simple terms, it's all about choosing the right number to lead each row, which we call the "pivot". These pivot points guide the transformations we perform on the matrix to help simplify our equations.

• In each step, you look for the strongest (usually the largest in absolute value) leading coefficient in the column you're working with. This is your pivot.
• Once you've found the pivot, you might need to swap it to the top of your working column. This ensures your calculations are stable and precise.
• In our initial example, the pivot in the first column (1st row with 4) was already in place, so no swaps were needed. Sometimes you're lucky like that!

Why is pivoting so important? Well, it helps manage errors, especially when working with decimals or fractions. Proper pivoting leads to fewer mistakes and a smoother path to row reduction. It's like building a strong foundation when constructing a house!

Reduced Row Echelon Form

Getting a matrix into Reduced Row Echelon Form (RREF) is like reaching the summit when using Gaussian Elimination. This form provides a clean and straightforward pathway to solutions of linear equations. It helps you see relationships between variables more clearly.
Here's what defines RREF:

• Every leading entry (pivot) in a row is 1.
• Each pivot is the only non-zero entry in its column.
• The pivots appear sequentially from left to right as you move down the rows.
• Any rows of all zeroes are at the bottom of the matrix.

From our example, through systematic steps, we manipulated the initial matrix into a nice, triangular form. This means each variable in the system can be expressed cleanly.
Once in this form, solving the system is just about reading off the solutions from the matrix. Think of RREF as a perfectly organized tool rack where every component is neatly in its place, making the task at hand more manageable.

Matrix Operations

Matrix operations are the toolkit for Gaussian Elimination. These operations - including row swaps, scaling rows, and adding/subtracting rows - transform a system of linear equations into a more solvable form by reshaping the associated matrix.
**The Main Matrix Operations Involved:**

• Row Swapping: Change the position of two rows if it helps to get a larger pivot on top. This was unnecessary in our exercise but could be vital in other situations.
• Scaling Rows: Multiply or divide every term in a row by a non-zero constant to achieve the desired pivot. This ensures that leading coefficients (pivots) become 1.
• Row Addition/Subtraction: Add or subtract multiples of one row from another to eliminate unwanted entries and create zeros in specific positions.

In our Gaussian Elimination task, these operations were performed to systematically zero out entries below each pivot and reduce the matrix to its rudimentary form. Engaging in these matrix maneuvers is like sculpting; each operation is a chisel stroke towards achieving a polished and clean result.