What elementary row operations do the following matrices...

What elementary row operations do the following matrices represent? What size matrices do they apply to? (a) $\left(\begin{array}{rr}1 & -2 \\ 0 & 1\end{array}\right)$, (b) $\left(\begin{array}{ll}1 & 0 \\ 7 & 1\end{array}\right)$, (c) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & -5 \\ 0 & 0 & 1\end{array}\right)$, (d) $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & 1\end{array}\right)$, (e) $\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right)$.

What elementary row operations do the following matrices represent? What size matrices do they apply to?
(a) $\left(\begin{array}{rr}1 & -2 \\ 0 & 1\end{array}\right)$,
(b) $\left(\begin{array}{ll}1 & 0 \\ 7 & 1\end{array}\right)$,
(c) $\left(\begin{array}{rrr}1 & 0 & 0 \\ 0 & 1 & -5 \\ 0 & 0 & 1\end{array}\right)$,
(d) $\left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ \frac{1}{2} & 0 & 1\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{array}\right)$.

Key Concepts

Elementary Row Operations

Elementary row operations are fundamental actions performed on the rows of a matrix to simplify it, solve linear systems, or compute its inverse. They include operations such as swapping two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. These operations are key tools in methods like Gaussian elimination.

Elementary Matrices

Elementary matrices are special matrices obtained by applying a single elementary row operation to an identity matrix. When an elementary matrix is multiplied on the left of another matrix, it performs the corresponding row operation. This concept is central because it ties the abstract idea of a row operation to a concrete matrix representation, and helps in understanding matrix factorization and inversion.

Types of Elementary Row Operations

There are three primary types of elementary row operations: row swapping (interchanging two rows), row scaling (multiplying a row by a nonzero scalar), and row addition (replacing a row by itself plus a multiple of another row). Each type of operation has a corresponding elementary matrix, and recognizing these types is crucial for systematically transforming matrices during algorithmic procedures.

Matrix Size Considerations

Elementary matrices are always square matrices, with their size determined by the number of rows in the matrix that the operation is to be applied to. The size of an elementary matrix indicates that it operates on matrices with a compatible number of rows, meaning an n×n elementary matrix can be used to perform row operations on any matrix with n rows, irrespective of the number of columns.

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