Question

Find the rank of each of the following matrices. $$\left(\begin{array}{lll} 1 & 1 & 2 \\ 2 & 4 & 6 \\ 3 & 2 & 5 \end{array}\right)$$

Short answer

The rank of the matrix is 2.

Step-by-step solution

- Write the matrix

Start with the given matrix: \[ A = \begin{pmatrix} 1 & 1 & 2 \ 2 & 4 & 6 \ 3 & 2 & 5 \ \end{pmatrix} \]

- Begin row reduction

Use row operations to simplify the matrix. First, subtract 2 times the first row from the second row: \ \[ R_2 = R_2 - 2R_1 \rightarrow \begin{pmatrix} 1 & 1 & 2 \ 0 & 2 & 2 \ 3 & 2 & 5 \ \end{pmatrix} \]

- Continue row reduction

Next, subtract 3 times the first row from the third row: \ \[ R_3 = R_3 - 3R_1 \rightarrow \begin{pmatrix} 1 & 1 & 2 \ 0 & 2 & 2 \ 0 & -1 & -1 \ \end{pmatrix} \]

- Simplify further

Divide the second row by 2: \ \[ R_2 = \frac{1}{2}R_2 \rightarrow \begin{pmatrix} 1 & 1 & 2 \ 0 & 1 & 1 \ 0 & -1 & -1 \ \end{pmatrix} \]

- More row operations

Add the second row to the third row: \ \[ R_3 = R_3 + R_2 \rightarrow \begin{pmatrix} 1 & 1 & 2 \ 0 & 1 & 1 \ 0 & 0 & 0 \ \end{pmatrix} \]

- Identify the rank

Count the number of non-zero rows in the echelon form matrix. The reduced matrix is: \[ \begin{pmatrix} 1 & 1 & 2 \ 0 & 1 & 1 \ 0 & 0 & 0 \ \end{pmatrix} \] There are 2 non-zero rows.

Key concepts

Row Reduction

Row reduction, also known as Gaussian elimination, is a method used to simplify matrices. This process involves a series of operations performed on the rows of a matrix to transform it into a simpler form. These operations include row swapping, scaling rows, and adding or subtracting rows. The aim is to convert the matrix into an echelon form, from which we can easily determine the rank or solve linear equations.

Specifically, the steps generally involve:

• Choosing a pivot element (non-zero entry) in the leading position.
• Making the pivot entry one by scaling the row.
• Eliminating the other entries in the pivot column by adding or subtracting multiples of the pivot row.

In this exercise, we started by subtracting multiples of the first row to zero out elements below the pivot. By repeating this process for subsequent rows, we gradually clarified the structure of the matrix.

Echelon Form

Echelon form of a matrix, particularly row echelon form (REF), is a simplified, structured state of a matrix achieved through row reduction. A matrix is said to be in row echelon form if it satisfies the following conditions:

• All non-zero rows are above any rows of all zeros.
• The leading entry (first non-zero number from the left) of each non-zero row is strictly to the right of the leading entry of the row above it.
• The leading entry in any non-zero row is 1 (this condition is part of the reduced row echelon form, a further simplified version).

By transforming a matrix to its echelon form, we can easily identify important properties. In our example, the operations simplified the matrix into two non-zero rows and one zero row, helping us determine that the rank of the matrix is 2.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, matrices, and linear transformations. It's a fundamental part of modern mathematics and has applications in various fields, including physics, computer science, engineering, and economics. One key concept in linear algebra is the rank of a matrix. The rank of a matrix is the maximum number of linearly independent row or column vectors in the matrix. It provides insight into the matrix's properties, such as its invertibility and the solutions to linear equations.

In our exercise, by using row reduction to convert the matrix to echelon form, we counted the non-zero rows to find the rank. This demonstrates the connection between simplified forms of matrices and the broader concepts in linear algebra. Understanding and determining matrix rank allows students to tackle complex problems involving systems of linear equations and vector spaces.