Problem 34 - 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. \(\left(\begin{array}{llll}1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\\ 0 & 0 & 0 & 1\end{array}\right)\)

Step-by-Step Solution

Answer

Yes, it is an elementary matrix.

Step 1: Understand the Concept of an Elementary Matrix

An elementary matrix is derived from the identity matrix by performing exactly one row operation. Row operations include row swapping, row multiplication, or row addition.

Step 2: Examine the Given Matrix

The matrix given is \[\left(\begin{array}{cccc}1 & 0 & 0 & 1 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1\end{array}\right)\]This matrix needs to be compared to the identity matrix of the same size.

Step 3: Compare to the Identity Matrix

The identity matrix for a 4x4 matrix is given by:\[I = \left(\begin{array}{cccc}1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1\end{array}\right)\]Comparing this with the given matrix, we notice that the first row has been changed from \((1, 0, 0, 0)\) to \((1, 0, 0, 1)\).

Step 4: Identify the Row Operation

The row operation performed is adding a multiple of a row to another row. In this case, 1 times the 4th row has been added to the 1st row of the identity matrix, changing it to \((1, 0, 0, 1)\). No other rows are affected.

Step 5: Conclusion

Since only one row operation (adding a multiple of one row to another row of the identity matrix) was performed to obtain the given matrix, it qualifies as an elementary matrix.

Key Concepts

Row Operations

When studying matrices, row operations are quite fundamental. They are used to transform matrices and are essential in many matrix-related algorithms. Row operations fall into three main categories:

• Row swapping: This is a basic operation where two rows in the matrix are interchanged. This doesn't alter the determinant in specific types of matrices but changes the matrix structure.
• Row multiplication: In this operation, a row is multiplied by a non-zero constant. It changes the scaling of that particular row but doesn't change the solution set of linear equations represented by the matrix.
• Row addition: Here, a multiple of one row is added to another row. This is exactly the operation seen in the discussed elementary matrix. It modifies one row by adding a weighted amount of another row, enhancing our ability to eliminate variables in systems of equations.

Row operations are reversible, meaning any change can be undone with another row operation. This is crucial in solving linear equations using methods like Gaussian elimination.

Identity Matrix

The identity matrix is a special kind of square matrix that has 1s on its main diagonal and 0s elsewhere. For any matrix \(A\), multiplying by the identity matrix results in \(A\) itself:
\[ A \cdot I = A \quad \text{and} \quad I \cdot A = A \]
This is why it is called the identity - it does not alter the matrix it multiplies. It acts as the multiplicative identity in matrix algebra, similar to how multiplying any number by 1 gives us the same number.
The identity matrix is critical for defining elementary matrices, as an elementary matrix is formed from an identity matrix by applying a single row operation. These simple transformations teach us how different matrix operations impact equations, making the identity matrix a foundational element in linear algebra.
Think of it as a baseline or starting template for creating other matrices through row operations.

Matrix Addition

Matrix addition is an operation that allows us to add two matrices together by adding their corresponding elements. For matrices to be added, they must have the same dimensions. The process is straightforward but follows specific rules:

• Each element in the resulting matrix is the sum of the elements at the same position in the matrices being added.
• For example, if you add two 2x2 matrices \(\begin{pmatrix} a & b \ c & d \end{pmatrix}\) and \(\begin{pmatrix} e & f \ g & h \end{pmatrix}\), the result will be \(\begin{pmatrix} a+e & b+f \ c+g & d+h \end{pmatrix}\).

Matrix addition is useful for solving linear systems and comes into play when dealing with more complex matrix operations. While straightforward, it's crucial for understanding more advanced matrix algebra concepts, as it forms the basis for many operations.
Matrix addition keeps the algebra consistent, offering a way to perform operations that mirror basic arithmetic but within the matrix structure.