Question

The rank of matrix \(\left[\begin{array}{rrrr}2 & -1 & 3 & 1 \\ 1 & 4 & -2 & 1 \\\ 5 & 2 & 4 & 3\end{array}\right]\) is

Short answer

The rank of the matrix is 3.

Step-by-step solution

Write the Augmented Matrix

Start by writing the matrix given in the problem as an augmented matrix. The matrix is a 3x4 matrix:\[\begin{bmatrix} 2 & -1 & 3 & 1 \ 1 & 4 & -2 & 1 \ 5 & 2 & 4 & 3 \end{bmatrix}\]

Perform Row Reduction to Echelon Form

Use elementary row operations to convert the matrix into its row echelon form (REF). Start by making zeros below the first element in the first column of the matrix. For instance, subtract half of the first row from the second row, and subtract 5x the first row from the third row. Continue row operations until all non-zero rows have leading ones and all zeros are located beneath these leading ones.

Check Number of Non-Zero Rows

Once your matrix is in row echelon form, count the number of non-zero rows. The number of these rows will determine the rank of the matrix.

Conclude the Rank

After performing all row operations, the rows that contain at least one non-zero element determine the rank of the matrix. Ensure that you correctly count these rows to determine the final rank.

Key concepts

Understanding Row Reduction

Row reduction is a process used extensively in linear algebra to simplify matrices, helping to find their rank. The process involves applying a series of operations to transform a given matrix into a simpler form. The goal is to make it easier to solve linear equations or analyze the matrix itself.

When performing row reduction, you aim to create zeros below the leading coefficients (also called pivots) in each column. A leading coefficient is the first non-zero number from the left in a non-zero row. By systematically making zeros below these pivots, you simplify the matrix.

Row reduction is often the first step in analyzing a matrix and is essential for other processes, like determining the inverse of a matrix, if it exists. It is vital for students to become comfortable with these operations, as they also lay the groundwork for understanding more complex algebraic structures.

Row Echelon Form Simplified

The row echelon form (REF) of a matrix is a stage in the row reduction process where each row begins with a leading coefficient of 1, and all the zeros are below these leading coefficients. This form offers a simplified version of the original matrix and makes it easier to understand the linear relationships between the columns.

Characteristics of the row echelon form include:

• All nonzero rows are above any rows of all zeroes.
• The leading coefficient of a non-zero row is always to the right of the leading coefficient of the row above it.
• The leading coefficient in each non-zero row is 1.

Converting a matrix to its REF helps identify pivot positions, which are critical to calculating the matrix's rank, determining its solutions, and revealing other properties.

The Role of Elementary Row Operations

Elementary row operations are the tools needed to change a matrix into a form like the row echelon form. There are three types of elementary row operations:

• Swapping two rows.
• Multiplying a row by a nonzero scalar.
• Adding or subtracting a multiple of one row to another row.

These operations are pivotal because they do not change the solution set of the linear equations represented by the matrix. Instead, they help us reveal the matrix's characteristics, such as its rank. They allow us to strategically place zeros in certain positions, streamline solving systems of equations, or find the inverse of the matrix if applicable.

Embracing these operations enables an understanding of the structure and inherent characteristics of a matrix.

The Purpose of the Augmented Matrix

In solving systems of linear equations, an augmented matrix is a compact and efficient way to represent both the coefficients and constants of the equations. An augmented matrix brings together the data in an accessible layout that simplifies many linear algebra procedures.

When constructed, the augmented matrix combines the coefficient matrix and the constants matrix, separated by a vertical line to help differentiate them. By converting this matrix to row echelon form using row operations, we can quickly determine solutions to the system or investigate properties such as rank. The augmented matrix is foundational in matrix algebra, offering clarity and structure to complex algebraic expressions.