Problem 33 - Advanced Engineering Mathematics

Question

An elementary matrix \(\mathbf{E}\) is one obtained by performing a single row operation on the identity matrix I. Verify that the given matrix is an elementary matrix. \(\left(\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & c & 1\end{array}\right)\)

Step-by-Step Solution

Answer

Yes, this is an elementary matrix due to a valid row addition operation.

Step 1: Identify the Identity Matrix

The 3x3 identity matrix \( \mathbf{I} \) is the matrix:\[\begin{pmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1\end{pmatrix}\]This is the starting point for the elementary matrix, as elementary matrices are formed by performing a single row operation on the identity matrix.

Step 2: Recognize the Row Operation

For the given matrix \(\begin{pmatrix}1 & 0 & 0 \0 & 1 & 0 \0 & c & 1\end{pmatrix}\), identify the row operation applied to the identity matrix. The third row of the identity matrix has been altered to \( [0, c, 1] \), indicating that a multiple of the second row has been added to the third row (a row addition operation).

Step 3: Verify the Change Matches an Elementary Operation

In the given matrix, adding \(c\) times the second row to the third row transforms \( \begin{pmatrix} 0 & 0 & 1 \end{pmatrix} \) to \( \begin{pmatrix} 0 & c & 1 \end{pmatrix} \). This is a valid elementary row operation corresponding to adding a multiple of one row to another row, confirming this matrix as an elementary matrix.

Key Concepts

Identity Matrix

An identity matrix plays a crucial role in matrix algebra. It is essentially the matrix equivalent of the number one in basic arithmetic. When we multiply any matrix by an identity matrix, it remains unchanged. The identity matrix is always a square matrix, meaning it has the same number of rows and columns. The defining feature of an identity matrix is that all the diagonal elements are ones, while all the other elements are zeroes. For example, a 3x3 identity matrix looks like this:

• Its elements on the diagonal—1, 2, and 3—are all 1s.
• All other elements are 0s.

When dealing with elementary matrices, you'll often start with an identity matrix. This is because an elementary matrix is derived by performing a single row operation on an identity matrix. Understanding the structure of an identity matrix is crucial for recognizing these operations.

Row Operations

In the world of matrices, row operations are transformations that can alter matrix rows without disrupting the core relationship between entire matrices. There are three main types of row operations:

• Row Swapping: This is when you exchange two rows with each other.
• Row Multiplication: This involves multiplying a row by a non-zero scalar.
• Row Addition: Here, you add a multiple of one row to another row.

In our given problem, the matrix \[\begin{pmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & c & 1\end{pmatrix}\]underwent a row addition operation. Specifically, a multiple of the second row was added to the third row. You can spot these changes by comparing the matrix to the identity matrix: if one row has been altered, then a row operation has taken place. This understanding is crucial because each row operation results in an elementary matrix, which can be used in solving system of linear equations or inverting matrices.

Matrix Verification

Verifying that a matrix is elementary involves ensuring it adheres to the effects of elementary row operations performed on an identity matrix. To verify an elementary matrix, follow these steps:

• Start with an identity matrix of the same size and observe the changes made to it.
• Each change should correspond to a proper elementary row operation—either a row swap, row multiplication by a nonzero number, or row addition.
• For our exercise, the matrix\[\begin{pmatrix}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & c & 1\end{pmatrix}\]was derived from the identity matrix by altering only one row. Specifically, it added a multiple of the second row to the third row—making it elementary.

By confirming that only one row operation has taken place and that it matches one of the listed operations, the matrix can be verified as an elementary matrix. This verification process is essential for understanding linear transformations and matrix operations.