Question

Find the rank of each of the following matrices. $$\left(\begin{array}{cccc} 2 & -3 & 5 & 3 \\ 4 & -1 & 1 & 1 \\ 3 & -2 & 3 & 4 \end{array}\right)$$

Short answer

The rank of the matrix is 3.

Step-by-step solution

- Set Up the Matrix

Start with the matrix provided: \[ \left(\begin{array}{cccc} 2 & -3 & 5 & 3 \ 4 & -1 & 1 & 1 \ 3 & -2 & 3 & 4 \end{array}\right) \]

- Row Reduce to Echelon Form

Use row operations to convert the matrix to row echelon form. Start by making the element in the first column, second row zero by using row operation \(R2 = R2 - 2R1\) and \(R3 = R3 - \frac{3}{2}R1\): \[ \left( \begin{array}{cccc} 2 & -3 & 5 & 3 \ 0 & 5 & -9 & -5 \ 0 & \frac{5}{2} & \frac{-9}{2} & \frac{5}{2} \end{array} \right) \] Next, make the element in the second column, third row zero by doing \(R3 = R3 - \frac{1}{2}R2\): \[ \left( \begin{array}{cccc} 2 & -3 & 5 & 3 \ 0 & 5 & -9 & -5 \ 0 & 0 & 0 & 5 \end{array} \right) \]

- Identify Pivot Columns

Examine the row-reduced matrix to find pivot columns. A pivot column is a column that contains the leading entry (first non-zero entry) of a row. For the matrix after row reduction: \[ \left( \begin{array}{cccc} 2 & -3 & 5 & 3 \ 0 & 5 & -9 & -5 \ 0 & 0 & 0 & 5 \end{array} \right) \] the first, second, and fourth columns are pivot columns.

- Determine the Rank

The rank of the matrix is defined as the number of pivot columns. From the matrix, there are 3 pivot columns (the first, second, and fourth columns). Therefore, the rank of the matrix is 3.

Key concepts

Row Reduction

Row reduction is a key technique in linear algebra used for simplifying matrices. It involves a series of row operations to transform a given matrix into a more convenient form, such as row echelon form or reduced row echelon form. These operations do not change the fundamental properties of the matrix.

There are three main types of row operations:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting a multiple of one row to another

To start, we take the original matrix and aim to create zeros below each leading 1 (pivot) in its columns. For instance, in the provided problem, we first nullified the element in the first column, second row by applying the row operation \( R2 = R2 - 2R1 \).

By strategically applying these operations, one can simplify the matrix to reveal crucial information such as rank and the number of pivot columns.

Echelon Form

The echelon form of a matrix is an upper triangular form where each leading entry (or pivot) in a row is to the right of the leading entry in the row above. It makes it easier to solve systems of linear equations or identify key matrix properties.

To achieve row echelon form (REF), we use row reduction techniques:

• Convert the leftmost non-zero entry in each row to 1 (pivoting)
• Make all elements below the pivots zero (using row operations)
• Ensure each pivot is to the right of the pivot in the row above

In our example, after performing the necessary row operations, we obtain the matrix:
\[ \left( \begin{array}{cccc} 2 & -3 & 5 & 3 \ 0 & 5 & -9 & -5 \ 0 & 0 & 0 & 5 \end{array} \right) \]

This form gives us clear visibility of the pivot positions and allows us to proceed directly to identifying the number of pivot columns.

Pivot Columns

Pivot columns in a matrix are columns that contain the first non-zero entry (or pivot) of each row in row echelon form. Identifying pivot columns is essential in determining the rank of a matrix.

After transforming the given matrix to echelon form, we look for the columns where the first nonzero entries appear. These entries are known as pivots. For the example matrix transformed to:
\[ \left( \begin{array}{cccc} 2 & -3 & 5 & 3 \ 0 & 5 & -9 & -5 \ 0 & 0 & 0 & 5 \end{array} \right) \]

The first, second, and fourth columns contain the pivots.

The number of pivot columns directly corresponds to the rank of the matrix. In this scenario, we have three pivot columns, which tells us that the rank of the matrix is 3.

Understanding pivot columns not only helps in finding the rank but also in solving linear equations and understanding the span and linear independence of a set of vectors.