(a) Suppose A̅ is obtained from A by applying an elementary...

(a) Suppose $\bar{A}$ is obtained from $A$ by applying an elementary row operation. Let $C=A B$, where $B$ is any matrix of the appropriate size. Explain why $\bar{C}=\bar{A} B$ can be obtained by applying the same elementary row operation to $C$. (b) Illustrate by adding -2 times the first row to the third row of $A=\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & -3 & 2 \\ 0 & 1 & -4\end{array}\right)$ and then multiplying the result on the right by $B=\left(\begin{array}{rr}1 & -2 \\ 3 & 0 \\ -1 & 1\end{array}\right)$. Check that the resulting matrix is the same as first multiplying $A B$ and then applying the same row operation to the product matrix.

Key Concepts

Elementary Row Operations

Elementary row operations include the basic manipulations applied to the rows of a matrix—swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. These operations are foundational in linear algebra for solving systems of equations, finding inverses, and performing row reduction. Their consistent behavior and properties make them key tools in transforming matrices while preserving certain invariant characteristics of the system.

Elementary Matrices

Each elementary row operation can be represented by an elementary matrix, which is obtained by performing that row operation on an identity matrix. When an elementary matrix multiplies another matrix from the left, it has the same effect as performing the corresponding row operation directly on that matrix. This links the concept of matrix multiplication with structural transformations induced by elementary row operations.

Matrix Multiplication Compatibility

Matrix multiplication is compatible with row operations because the act of left-multiplying a matrix by an elementary matrix (performing a row operation) distributes over the product of matrices. In other words, if a row operation is applied to a matrix A to yield A-bar, then multiplying A-bar by B results in the same modified matrix as multiplying A by B and then performing the row operation on the product. This property underpins many techniques in linear algebra, such as simplifying systems or matrices for solving equations.

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