Question

Find the rank of each of the following matrices. $$\left(\begin{array}{rrrr} 1 & 0 & 1 & 0 \\ -1 & -2 & -1 & 0 \\ 2 & 2 & 5 & 3 \\ 2 & 4 & 8 & 6 \end{array}\right)$$

Short answer

The rank of the matrix is 3.

Step-by-step solution

Write Down the Given Matrix

The given matrix is: \[ A = \begin{pmatrix} 1 & 0 & 1 & 0 \ -1 & -2 & -1 & 0 \ 2 & 2 & 5 & 3 \ 2 & 4 & 8 & 6 \end{pmatrix} \]

Perform Row Operations to Convert Matrix to Row Echelon Form (REF)

We want to simplify the matrix by performing elementary row operations. First, set the matrix in the '[' ']' form and perform the following operations:1. Add Row 2 (R2) to Row 1 (R1):\[R1 + R2 \rightarrow R2\] \[\begin{pmatrix} 1 & 0 & 1 & 0 \ 0 & -2 & 0 & 0 \ 2 & 2 & 5 & 3 \ 2 & 4 & 8 & 6 \end{pmatrix}\]2. Subtract 2 times R1 from R3:\[R3 - 2R1 \rightarrow R3\]\[\begin{pmatrix} 1 & 0 & 1 & 0 \ 0 & -2 & 0 & 0 \ 0 & 2 & 3 & 3 \ 2 & 4 & 8 & 6 \end{pmatrix}\]3. Subtract 2 times R1 from R4:\[R4 - 2R1 \rightarrow R4\]\[\begin{pmatrix} 1 & 0 & 1 & 0 \ 0 & -2 & 0 & 0 \ 0 & 2 & 3 & 3 \ 0 & 4 & 6 & 6 \end{pmatrix}\]4. Divide Row 2 (R2) by -2:\[ \frac{R2}{-2} \rightarrow R2\]\[\begin{pmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 2 & 3 & 3 \ 0 & 4 & 6 & 6 \end{pmatrix}\]5. Subtract 2 times Row 2 from R3:\[ R3 - 2R2 \rightarrow R3\]\[\begin{pmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 3 \ 0 & 4 & 6 & 6 \end{pmatrix}\]6. Subtract 4 times Row 2 from R4:\[ R4 - 4R2 \rightarrow R4\]\[\begin{pmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 3 \ 0 & 0 & 6 & 6 \end{pmatrix}\]7. Subtract 2 times Row 3 from Row 4\[ R4 - 2R3 \rightarrow R4\]\[\begin{pmatrix} 1 & 0 & 1 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 3 \ 0 & 0 & 0 & 0 \end{pmatrix}\]

Determine the Rank of the Matrix

Count the number of non-zero rows in the row echelon form of the matrix. The rows are:1. \( \begin{pmatrix} 1 & 0 & 1 & 0 \end{pmatrix} \)2. \( \begin{pmatrix} 0 & 1 & 0 & 0 \end{pmatrix} \)3. \( \begin{pmatrix} 0 & 0 & 3 & 3 \end{pmatrix} \)There are 3 non-zero rows.Hence, the rank of the matrix is 3.

Key concepts

row operations

Row operations are techniques used to manipulate the rows of a matrix. They help us simplify matrices into forms that make calculations easier. There are three main types of row operations: swapping two rows, multiplying a row by a non-zero constant, and adding or subtracting a multiple of one row to another. These operations are essential for converting a matrix into its row echelon form (REF).

They are crucial in various matrix calculations like finding the rank, solving linear equations, and determining matrix invertibility. Understanding and mastering row operations is foundational for anyone diving into linear algebra.

row echelon form

Row echelon form (REF) is a simplified version of a matrix where leading entries (first non-zero numbers) of each row move from left to right as you go down the matrix.
In this form, all rows consisting solely of zeros are at the bottom.

Achieving row echelon form makes it easier to analyze the matrix, particularly for determining the rank. The process involves using elementary row operations to create zeros below each leading entry. Once in REF, the matrix reveals the underlying structure, such as linearly independent rows, assisting in various computations in linear algebra.

elementary row operations

Elementary row operations are the basic tools for manipulating matrices. These are indispensable in converting a matrix into row echelon form. The three operations are:

• Row Swapping: Interchanging two rows of a matrix.
• Row Multiplication: Multiplying all elements of a row by a non-zero constant.
• Row Addition/Subtraction: Adding or subtracting a multiple of one row to another row.

These operations preserve the solutions of a system of linear equations and do not change the rank of the matrix. Understanding how and when to use these operations aids significantly in matrix manipulation and problem-solving.

linear algebra

Linear algebra is a branch of mathematics that explores vectors, vector spaces, linear transformations, and systems of linear equations. It's a critical area of study due to its vast applications in multiple fields such as engineering, physics, computer science, and economics.

Key concepts include matrices, determinants, eigenvalues, and eigenvectors. The rank of a matrix, for example, helps determine the solutions to systems of linear equations. Gaining a solid understanding of linear algebra equips you with tools for solving complex problems in both theoretical mathematics and practical applications.

In summary, linear algebra provides the framework to understand and work with multiple linear equations and transformations, forming the backbone of various scientific and engineering calculations.