Problem 12 - Differential Equations and Linear Algebra

Question

Determine \(A^{-1},\) if possible, using the Gauss-Jordan method. If \(A^{-1}\) exists, check your answer by verifying that \(A A^{-1}=I_{n}\) $$A=\left[\begin{array}{rrr} 4 & 2 & -13 \\ 2 & 1 & -7 \\ 3 & 2 & 4 \end{array}\right]$$

Step-by-Step Solution

Answer

The given matrix \(A\) is singular, so it does not have an inverse.

Step 1: Create the augmented matrix

First, we create the augmented matrix by placing matrix A on the left and an identity matrix on the right: \[ \left[\begin{array}{rrr|rrr} 4 & 2 & -13 & 1 & 0 & 0 \\ 2 & 1 & -7 & 0 & 1 & 0 \\ 3 & 2 & 4 & 0 & 0 & 1 \end{array}\right] \]

Step 2: Perform row operations to get the left side as identity matrix

Apply row operations to transform the left side into an identity matrix. We will first focus on obtaining a 1 in the top left corner of the matrix (1,1) position. 1. Divide Row1 by 4 to get 1 in (1,1) position: \[ \left[\begin{array}{rrr|rrr} 1 & \frac{1}{2} & -\frac{13}{4} & \frac{1}{4} & 0 & 0 \\ 2 & 1 & -7 & 0 & 1 & 0 \\ 3 & 2 & 4 & 0 & 0 & 1 \end{array}\right] \] 2. Eliminate the 2 in Row2 and 3 in Row3 from (2,1) and (3,1) positions: - Subtract 2 * Row1 from Row2 - Subtract 3 * Row1 from Row3 \[ \left[\begin{array}{rrr|rrr} 1 & \frac{1}{2} & -\frac{13}{4} & \frac{1}{4} & 0 & 0 \\ 0 & 0 & 0 & -\frac{1}{2} & 1 & 0 \\ 0 & 1 & 15 & -\frac{3}{4} & 0 & 1 \end{array}\right] \] Now, we can see that the matrix A is singular because it has a row of zeros. This means that A has no inverse.

Step 3: Conclusion

The given matrix \(A\) is singular, so it does not have an inverse. There is no need to verify the result, as there is no inverse to check.

Key Concepts

Gauss-Jordan elimination

The Gauss-Jordan elimination method is a process used in linear algebra to find the inverse of a matrix, if it exists, or to solve systems of linear equations. This method involves performing a series of row operations on an augmented matrix until the left portion of the matrix is transformed into the identity matrix.

These row operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another. Successfully carrying out the Gauss-Jordan elimination leads to a modified right portion that is the inverse of the original matrix. It’s important to note that the inverse only exists if the matrix is non-singular, meaning it must not contain any rows of zeros once the left side has been converted into the identity matrix.

Augmented matrix

An augmented matrix is essentially the fusion of two matrices side-by-side, and it is extensively used in methods like Gauss-Jordan elimination. To start, you create this matrix by attaching the identity matrix to the right side of the original square matrix.

The identity matrix is a special matrix in linear algebra, characterized by ones on the diagonal and zeros elsewhere. By crafting this combination, you can perform row operations on the augmented matrix to change the original matrix part (the left side) into the identity matrix. If this process is possible, the right side undergoes the same row operations and becomes the inverse of the original matrix.

Row operations

Row operations are fundamental tools in various matrix manipulation techniques such as the Gauss-Jordan elimination. There are three types of row operations:

• Row swapping: Interchanging two rows of a matrix.
• Row multiplication: Multiplying all entries of a row by a non-zero scalar.
• Row addition: Adding or subtracting one row to another after multiplying it by a scalar if needed.

These operations are employed to systematically simplify the matrix into a desired form, such as converting the left side of an augmented matrix into an identity matrix in order to discover the inverse of the original matrix.

Singular matrix

A singular matrix is a square matrix that does not have an inverse. One of the key indicators of a matrix being singular is the presence of a row or a column of all zeros after performing row operations, as seen in the given exercise with the matrix A.

This condition implies that the rows of the matrix are linearly dependent. In more practical terms, it means there's no unique solution to the corresponding system of linear equations. Identifying a singular matrix early in calculations can save time, as it is not necessary to complete the Gauss-Jordan elimination process if the goal is to find an inverse that does not exist.