Problem 33 - Differential Equations and Linear Algebra

Question

(a) find a row-echelon form of the given matrix \(A,\) (b) determine rank \((A),\) and (c) use the GaussJordan Technique to determine the inverse of \(A,\) if it exists. $$A=\left[\begin{array}{rr}2 & -7 \\ -4 & 14\end{array}\right].$$

Step-by-Step Solution

Answer

The row-echelon form of matrix \(A\) is: \( A=\left[\begin{array}{rr}2 & -7 \\\ 0 & 0\end{array}\right] \) The rank of matrix \(A\) is 1, and the matrix \(A\) does not have an inverse.

Step 1: Finding Row-echelon form of A

Firstly, we need to find the row-echelon form of matrix \(A\) which is given by: \( A=\left[\begin{array}{rr}2 & -7 \\\ -4 & 14\end{array}\right] \) To do this, we perform the following row operation: \( R_{2} \rightarrow R_{2} + 2R_{1} \) The resulting matrix is: \( A=\left[\begin{array}{rr}2 & -7 \\\ 0 & 0\end{array}\right] \) Matrix \(A\) is now in row-echelon form.

Step 2: Determining Rank of A

The rank of a matrix is the number of linearly independent rows. In our row-echelon form, we have only one nonzero row. So the rank is 1, rank(A) = 1.

Step 3: Finding the Inverse of A using Gauss-Jordan Technique

To find the inverse of a matrix using the Gauss-Jordan Technique, we need to augment our matrix \(A\) with the identity matrix \(I\) of the same order. Since \(A\) is of order 2x2, we augment it with a 2x2 identity matrix: \( \left[\begin{array}{rr|rr}2 & -7 & 1 & 0 \\\ -4 & 14 & 0 & 1\end{array}\right] \) Now, we try to bring our augmented matrix to the reduced row-echelon form, i.e., the identity matrix on the left side. If we can get the identity matrix on the left side, then the matrix on the right will be the inverse of our original matrix \(A\). Let's apply the row operation that would lead to the reduced row-echelon form. \( R_{2} \rightarrow R_{2} + 2R_{1} \) The resulting matrix is: \( \left[\begin{array}{rr|rr}2 & -7 & 1 & 0 \\\ 0 & 0 & 1 & 0\end{array}\right] \) Notice that, we cannot form an identity matrix on the left side as there is one row of zeros. This means the matrix \(A\) is not invertible and does not have an inverse. In conclusion, the row-echelon form of matrix \(A\) is: \( A=\left[\begin{array}{rr}2 & -7 \\\ 0 & 0\end{array}\right] \) The rank of matrix \(A\) is 1, and the matrix \(A\) does not have an inverse.

Key Concepts

Row-echelon Form

The concept of row-echelon form is pivotal in matrix operations as it simplifies the process of solving systems of linear equations. To convert a matrix to row-echelon form, we employ row operations, aiming to form a stair-stepped shape with leading coefficients of 1. In simpler terms, non-zero rows are positioned above rows of zeros, and every leading entry of a row is in a column to the right of the leading entry of the row above it.
For the given matrix \(A = \begin{bmatrix} 2 & -7 \ -4 & 14 \end{bmatrix}\), we used a single row operation \( R_{2} \rightarrow R_{2} + 2R_{1} \) to achieve the row-echelon form.
Here is why this operation transformed the matrix correctly:

• The operation aimed to eliminate the leading value in the second row, transforming \(-4\) to \(0\).
• The resulting matrix \(\begin{bmatrix} 2 & -7 \ 0 & 0 \end{bmatrix}\) is in the correct row-echelon form as the second row consists entirely of zeros, leaving the first row with non-zero values.

The row-echelon form is essential as it paves the way for determining the rank and other matrix properties.

Matrix Rank

The rank of a matrix is a fundamental concept that provides insightful information about the solutions of a system of linear equations associated with the matrix. In essence, the rank of a matrix is determined by the number of linearly independent rows or columns. A matrix's rank tells us the dimension of the vector space spanned by its rows or columns.
For the matrix \(A\), now in row-echelon form \(\begin{bmatrix} 2 & -7 \ 0 & 0 \end{bmatrix}\), it is clear that there is only one non-zero row. Thus, the rank of matrix \(A\) is \(1\). This indicates that the vectors formed by the rows of the matrix cannot span more than one-dimensional space.
Key points about matrix rank:

• A higher rank means more independent rows, affecting the matrix's invertibility and the solutions to linear systems.
• If all rows or columns are linearly dependent, then the matrix has a lower rank, like our matrix \(A\).

Understanding matrix rank is crucial when performing tasks such as determining the feasibility of finding an inverse.

Gauss-Jordan Technique

The Gauss-Jordan Technique is an extension of Gaussian elimination and is used to further simplify matrices to the reduced row-echelon form. This technique is vital when computing the inverse of a matrix, given that the inverse exists.
Here's a simplified view of the Gauss-Jordan process applied to our exercise:

• Augment the original matrix with an identity matrix of the same dimensions. For \(A\), this was a \(2x2\) identity matrix.
• Apply row operations with the goal of obtaining the identity matrix on the left side of the vertical bar.

If we succeed, the right side of the augmented matrix gives the inverse. However, our matrix \( A \) with row-echelon form \(\begin{bmatrix} 2 & -7 \ 0 & 0 \end{bmatrix}\) failed to yield an identity matrix, indicated by a row of zeros remaining. This confirmed that \( A \) is not invertible.
The Gauss-Jordan Technique is substantial as it provides a systematic approach for matrix inversion but also highlights scenarios of non-invertibility when the rank is insufficient.