Problem 22 - Differential Equations and Linear Algebra

Question

Reduce the given matrix to reduced rowechelon form and hence determine the rank of each matrix. $$\left[\begin{array}{ccc} 3 & -3 & 6 \\ 2 & -2 & 4 \\ 6 & -6 & 12 \end{array}\right]$$.

Step-by-Step Solution

Answer

The reduced row echelon form of the given matrix is: \(\left[\begin{array}{ccc} 1 & -1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\) The rank of the matrix is 1, as there is only one linearly independent row.

Step 1: Make the first row's leading term 1

Divide the first row by 3: \(\left[\begin{array}{ccc} 1 & -1 & 2 \\ 2 & -2 & 4 \\ 6 & -6 & 12 \end{array}\right]\)

Step 2: Eliminate the first term in the second and third rows

Subtract 2 times the first row from the second row and subtract 6 times the first row from the third row: \(\left[\begin{array}{ccc} 1 & -1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\)

Step 3: Rearrange the rows if needed

The current order of rows is already correct, as the leading 1 is below the previous row's leading 1. Now, the matrix is in its reduced row echelon form: \(\left[\begin{array}{ccc} 1 & -1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\)

Step 4: Determining the rank

Now, we can determine the rank of the matrix by counting the number of linearly independent rows (rows that are not all 0). In this case, there is only one such row, so: Rank = 1

Key Concepts

Row Reduction

Row reduction, also known as Gaussian elimination, is a process used to simplify matrices and solve systems of linear equations. The goal of row reduction is to convert a matrix into its reduced row echelon form (RREF). This is done through a series of row operations which include:

• Swapping rows: Exchanging the positions of two rows.
• Multiplying a row by a nonzero constant: Scaling the entire row by a non-zero number.
• Adding or subtracting a multiple of one row to another row: This operation is used to eliminate unwanted elements in rows.

These operations help simplify the matrix to have leading ones (first non-zero number from the left) and zeros below these leading ones in their columns. The ultimate objective is to place zeros above and below each leading one, making the matrix simple to interpret and use in solving linear systems.
In our example, through row reduction, the given matrix is transformed such that non-essential rows become zero. This transformation helps us understand the properties and solutions related to the original matrix.

Matrix Rank

The rank of a matrix is a fundamental concept in linear algebra that reveals the dimension of the vector space spanned by the rows or columns of the matrix. To determine a matrix's rank, one must count the number of non-zero rows in its reduced row echelon form. Each non-zero row signifies a dimension of the solution space that the matrix represents.
In the provided example matrix, after row reduction, the matrix becomes:equation: \[\begin{bmatrix}1 & -1 & 2 \0 & 0 & 0 \0 & 0 & 0\end{bmatrix}\]Here, only the first row is non-zero, indicating that the matrix's rank is \(1\). This tells us that there is one linearly independent row in the matrix, which can be visualized as a single dimension in the vector space she matrix embodies.
Understanding the rank offers insights into the matrix's properties, such as whether systems of equations it represents have unique solutions, no solution, or infinitely many solutions.

Linear Independence

Linear independence is a key concept in determining the relationships between vectors, particularly in the context of row and column vectors in matrices. Vectors are considered linearly independent if no vector in the set can be written as a linear combination of the others. This means that no vector is redundant in spanning the space represented by the set.
In terms of a matrix, check for linear independence involves transforming the matrix into its reduced row echelon form and examining the rows. Rows that remain non-zero indicate linear independence among the original rows.

• In the original matrix given as:\[\begin{bmatrix}3 & -3 & 6 \2 & -2 & 4 \6 & -6 & 12\end{bmatrix}\]Row reduction reveals the matrix:\[\begin{bmatrix}1 & -1 & 2 \0 & 0 & 0 \0 & 0 & 0\end{bmatrix}\]
• Here, only the first row is non-zero, meaning it is the only row that is linearly independent.

Understanding linear independence helps in grasping the concept of span and basis of a vector space. In practical terms, it tells us that the remaining rows are derived from the first row, and thus, do not contribute to extending the dimensionality of the space.