Problem 12 - Differential Equations and Linear Algebra

Question

use elementary row operations to reduce the given matrix to row-echelon form, and hence determine the rank of each matrix. $$\left[\begin{array}{rrr} 2 & 1 & 4 \\ 2 & -3 & 4 \\ 3 & -2 & 6 \end{array}\right]$$.

Step-by-Step Solution

Answer

The row-echelon form of the given matrix is: \[ \left[\begin{array}{rrr} 2 & 0 & 4 \\\ 0 & 1 & 0 \\\ 0 & 0 & 0 \end{array}\right] \] Since the row-echelon form has 2 non-zero rows, the rank of the matrix is 2.

Step 1: Original matrix

Copy down the original matrix, which is: \[ \left[\begin{array}{rrr} 2 & 1 & 4 \\\ 2 & -3 & 4 \\\ 3 & -2 & 6 \end{array}\right] \]

Step 2: Eliminate the first entry of the second and third rows

We can perform the following row operations to eliminate the first entry of the second and the third rows: 1. Row2 = Row2 - Row1 2. Row3 = Row3 - 1.5 * Row1 Our new matrix will now become: \[ \left[\begin{array}{rrr} 2 & 1 & 4 \\\ 0 & -4 & 0 \\\ 0 & -3.5 & -0.0 \end{array}\right] \]

Step 3: Eliminate the second entry of the third row

Now, we'll perform the following row operations to eliminate the second entry of the third row: 1. Row2 = -1/4 * Row2 2. Row3 = Row3 - (7/8) * Row2 The new matrix will become: \[ \left[\begin{array}{rrr} 2 & 1 & 4 \\\ 0 & 1 & 0 \\\ 0 & 0 & 0 \end{array}\right] \]

Step 4: Obtain the row-echelon form

We will now perform the following row operation to obtain the row-echelon form: 1. Row1 = Row1 - Row2 The new matrix in row-echelon form will be: \[ \left[\begin{array}{rrr} 2 & 0 & 4 \\\ 0 & 1 & 0 \\\ 0 & 0 & 0 \end{array}\right] \]

Step 5: Determine the rank of the matrix

Now that we have the row-echelon form of the matrix, we can determine its rank. The rank of a matrix is the number of nonzero rows in its row-echelon form. In this case, we have 2 nonzero rows, so the rank of the given matrix is 2.

Key Concepts

Elementary Row Operations

In linear algebra, elementary row operations are used to manipulate matrices, with one of the main goals being to transform a given matrix into row-echelon form. These operations are fundamental when solving systems of linear equations, determining matrix rank, and many other applications. There are three types of elementary row operations:

• Row switching, where two rows of the matrix are interchanged.
• Row multiplication, where each entry in a row is multiplied by a nonzero scalar.
• Row addition, where one row is added to another after multiplying it by a scalar.

Each of these operations alters the matrix while preserving its row-equivalence, importantly keeping the solution set of the associated system of equations unchanged. In our example, we used row addition and row multiplication to simplify the matrix. It's also critical to recognize the order in which we apply these operations significantly affects the efficiency of the process and the simplicity of the resulting row-echelon form.

Matrix Rank

Matrix rank is a fundamental concept that reflects the number of dimensions spanned by the rows (or columns) of the matrix, often corresponding to the number of linearly independent rows (or columns). In simpler terms, the rank of a matrix can be seen as an indicator of how much information the matrix contains.

• A full rank matrix has a rank equal to the smaller of the number of rows or columns.
• A matrix with rank deficiency has fewer linearly independent rows or columns than the maximum possible and reflects dependencies within the system it represents.

To determine the rank, one typically reduces a matrix to row-echelon form using elementary row operations. Nonzero rows indicate linearly independent rows, thus in our textbook exercise, the given matrix has 2 nonzero rows after reaching row-echelon form, which means its rank is 2. This is a pivotal number for understanding the solutions to corresponding systems of equations and the properties of the matrix.

Linear Algebra

Linear algebra, a significant branch of mathematics, focuses on vector spaces and linear mappings between such spaces. It forms the basis for many areas of mathematics and applications in science and engineering, enabling the study and understanding of planes, spaces, and multiple dimensions. It underpins various algorithms, including those used in machine learning and computer graphics.

• It deals with linear equations and their representations through matrices and vector spaces.
• Concepts such as linear independence, span, basis, and dimension are crucial for dissecting vector spaces.
• Eigenvalues and eigenvectors provide insight into matrix transformations and system behaviours.

Linear algebra techniques like matrix manipulation, the one you've learned in the exercise, are not just academic; they are practical methods used by various professionals to solve real-world problems.