Question

Find the rank of each of the following matrices. $$\left(\begin{array}{cccc} 1 & 1 & 4 & 3 \\ 3 & 1 & 10 & 7 \\ 4 & 2 & 14 & 10 \\ 2 & 0 & 6 & 4 \end{array}\right)$$

Short answer

The rank of the given matrix is 2.

Step-by-step solution

Write Down the Matrix

Begin by writing down the given matrix for clarity:enclosing all matrix rows

Perform Row Operations

Perform elementary row operations (like row swapping, adding/subtracting rows or multiplying a row by a nonzero scalar) to transform the matrix into its Row Echelon Form (REF).

Simplify Row 1

Focus on row 1. Multiply row 1 by 3 and subtract row 2: enclosing the resulting matrix.

Simplify Row 2

Subtract 4 times row 1 from row 3 to simplify row 3. Continue simplifying by subtracting twice row 1 from row 4 so that the first element is 0 in row 2.

Simplify Row 3

Perform row operations to transform row 3 by removing the influence of row 2.

Obtain Row Echelon Form

Keep using elementary row operations until the matrix is in Row Echelon Form. The matrix now should have the form:enclosing the echelon form matrix

Count Non-Zero Rows

Count the number of non-zero rows in the Row Echelon Form. Each non-zero row represents a pivot element.

Determine the Rank

The rank of the matrix is the number of non-zero rows in the Row Echelon Form.

Key concepts

Row Echelon Form

The Row Echelon Form (REF) of a matrix is one of the simplest forms a matrix can be reduced to using elementary row operations. Think of REF as a standardized shape for matrices. In this form, the matrix will have zeros below each pivot element. The pivots are the first non-zero numbers in each row. Here are the key properties of REF:

• Every non-zero row has a leading 1, called a pivot element.
• Pivot elements of subsequent rows appear to the right of the pivot elements in the preceding rows.
• The rows consisting entirely of zeros, if any, are at the bottom.

Transforming a matrix to its REF helps in determining the rank, solving systems of linear equations, and simplifying other matrix-related problems. The matrix provided in the exercise can be transformed into its REF through a series of elementary row operations.

Elementary Row Operations

Elementary row operations are the basic tools used to transform matrices. These operations can change the structure of the matrix while maintaining its fundamental properties. There are three types of elementary row operations:

• Row Swapping: Exchanging two rows of the matrix.
• Row Multiplication: Multiplying all elements of a row by a non-zero scalar.
• Row Addition/Subtraction: Adding or subtracting the multiples of one row from another row.

In practice, these operations allow us to systematically reduce the given matrix to its Row Echelon Form (REF). For example, in the exercise, rows were adjusted through multiplication and subtraction to achieve the form. This powerful set of operations ensures the transformation is reversible and the properties of the original matrix (like rank) are preserved.

Pivot Element

A pivot element in a matrix is the first non-zero element in each row when a matrix is in Row Echelon Form (REF). Pivots are crucial as they identify the columns that contribute to the rank.
When performing elementary row operations, we aim to create zeros below each pivot to simplify the matrix progressively. Here are key aspects of pivot elements:

• Each pivot is the leading 1 in its row.
• Pivots move to the right as you move down the rows in REF.
• The location and number of pivots are pivotal (pun intended) in determining the rank.

For instance, in the exercise, through systematic row operations, we identify these pivot elements to count the number of linearly independent rows, helping in the rank determination.

Non-Zero Rows

Non-zero rows in a matrix are rows that have at least one element different from zero. Counting these rows in the Row Echelon Form (REF) directly determines the rank of the matrix.
When a matrix is reduced into its REF, non-zero rows indicate linearly independent rows. Let's recall the steps:

• Each non-zero row corresponds to a pivot element.
• There are no non-zero elements below the pivot in each column.
• Zero rows, if any, are moved to the bottom.

In the exercise, by transforming the matrix into REF, we count the non-zero rows to find the rank. The number of these non-zero rows (typically the number of pivot elements) gives the rank of the matrix, which reflects how much of the matrix is linearly independent and spans the vector space.