Elementary Matrices

Elementary matrices are matrices obtained by performing a single elementary row operation on an identity matrix. These matrices are fundamental in linear algebra, especially for understanding row operations, matrix inversion, and solving systems of linear equations.

There are three types of elementary row operations, each corresponding to a type of elementary matrix:

1. Row Switching: Interchanging two rows of a matrix.
2. Row Multiplication: Multiplying a row by a non-zero scalar.
3. Row Addition: Adding a multiple of one row to another row.

Examples and Solutions

Example 1: Row Switching

Consider the 3x3 identity matrix $I_3$:

1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ To create an elementary matrix that swaps the first and second rows, perform the row operation $$R_1 \leftrightarrow R_2$$: $$E_1 = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ This matrix $$E_1$$ is an elementary matrix that, when multiplied by another matrix, swaps the first and second rows of that matrix. #### Example 2: Row Multiplication Starting with $$I_3$$ again: $$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ To create an elementary matrix that multiplies the second row by 3, perform the row operation $$3R_2$$: $$E_2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ This matrix $$E_2$$ is an elementary matrix that, when multiplied by another matrix, multiplies the second row of that matrix by 3. #### Example 3: Row Addition Starting with $$I_3$$ again: $$I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ To create an elementary matrix that adds twice the first row to the second row, perform the row operation $$R_2 + 2R_1$$: $$E_3 = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ This matrix $$E_3$$ is an elementary matrix that, when multiplied by another matrix, adds twice the first row to the second row of that matrix. ### Summary Each elementary matrix corresponds to a specific row operation, and when you multiply any matrix by an elementary matrix, that row operation is applied to the matrix. These operations are essential in algorithms like Gaussian elimination. --- Would you like more details on how these matrices are used in practice, or do you have any questions? ### Related Questions: 1. How are elementary matrices used in Gaussian elimination? 2. Can an elementary matrix be inverted? If so, how? 3. What is the role of elementary matrices in finding the inverse of a matrix? 4. How do elementary matrices relate to row echelon form? 5. What is the significance of elementary matrices in linear transformations? ### Tip: When working with elementary matrices, remember that the order of multiplication matters. Always apply the elementary operations in the correct sequence to achieve the desired row transformations.