Problem 31 - 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. \(\left(\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right)\)

Step-by-Step Solution

Answer

Yes, the given matrix is an elementary matrix formed by swapping two rows of the identity matrix.

Step 1: Understand the Definition of an Elementary Matrix

An elementary matrix can be defined as a matrix obtained by performing a single row operation on an identity matrix. These operations include switching two rows, multiplying a row by a nonzero scalar, or adding a multiple of one row to another row.

Step 2: Identify the Base Identity Matrix

Start with the identity matrix I, which matches the size of the given matrix. For a 3x3 matrix, the identity matrix is:\[I = \left(\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right)\]

Step 3: Compare Given Matrix to Identity Matrix

The given matrix is:\[\mathbf{E} = \left(\begin{array}{lll}0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{array}\right)\]Compare this matrix to the identity matrix and identify what row operation was performed.

Step 4: Identify the Row Operation

Observe that the first and second rows of the identity matrix have been swapped, resulting in:\[\text{Switching } R_1 \text{ and } R_2: \left(\begin{array}{lll}0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 1 \end{array}\right)\]This is a valid row operation on the identity matrix to obtain an elementary matrix.

Step 5: Conclude the Verification

Since swapping two rows of an identity matrix creates an elementary matrix, and the given matrix matches the form resulting from this row swap, it confirms that the matrix is indeed an elementary matrix.

Key Concepts

Row Operations

Row operations are fundamental transformations usually applied in linear algebra. They modify matrices while maintaining solutions to linear equations. These operations are basic and include three main types:

• Row Switching: This is swapping two rows in a matrix. It doesn't affect the matrix's determinant's absolute value but may change its sign.


• Row Multiplication: Here, every element of a row is multiplied by a nonzero scalar, modifying the row's magnitude. This operation affects the determinant proportionally to the scalar used.


• Row Addition: In this, a row is replaced by the sum of itself and a multiple of another row. This helps in linear combination and elementary row reduction.

When applied to an identity matrix, row operations result in an elementary matrix. This serves as an integral part of many matrix algorithms, such as Gaussian elimination, allowing transformations that simplify solving systems of equations.

Identity Matrix

An identity matrix, often denoted as \( I \), is a special type of square matrix with ones on the main diagonal and zeros elsewhere. For example, a 3x3 identity matrix looks like this:\[I = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}\]Identity matrices serve as the multiplicative identity in matrix algebra. This means if you multiply any matrix \( A \) by \( I \), the result is always \( A \). It acts similarly to the number 1 in basic arithmetic.
The properties of identity matrices make them essential for matrix operations, such as finding inverses and working within linear transformations. They form the baseline template for creating elementary matrices when subjected to row operations. Understanding their role is crucial in verifying and creating transformations within matrices.

Matrix Verification

Matrix verification involves confirming specific properties or transformations have been applied correctly. To verify that a matrix is an elementary matrix, you should first identify its origin from an identity matrix. This verification checks if the matrix is derived from a single row operation. Here's how you verify it:

• Identify the resemblance: Start with the identity matrix of matching size and compare the structure with the given matrix.


• Spot the operation: Look for exact differences indicating one of the defined row operations, such as row swaps or scalar multiplications.


• Confirm operations: Ensure only one row operation differentiates the given matrix from the identity matrix. If so, you've successfully verified the matrix as an elementary matrix.

Verification requires careful comparison and understanding of row operations. This check is an important skill in matrix algebra, ensuring that transformations follow logical and mathematical rules.