Question

A matrix \(A\) is given below. In Exercises 16 20, a matrix \(B\) is given. Give the row operation that transforms \(A\) into \(B\). $$A=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 2 & 3\end{array}\right]$$ $$B=\left[\begin{array}{lll}1 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 2 & 2\end{array}\right]$$

Short answer

Subtract the first row from the third row: \( R_3 \leftarrow R_3 - R_1 \).

Step-by-step solution

Identify the Difference

Compare the matrices \(A\) and \(B\) to identify the row that needs to be transformed. The matrix \(B\) is different from \(A\) in the third row, where the element in the first column has changed from 1 to 0.

Determine the Operation Needed

To change the first element of the third row in matrix \(A\) from 1 to 0, we can perform a row operation. The appropriate operation is to subtract the first row from the third row: \( R_3 \leftarrow R_3 - R_1 \).

Verify the Operation

Apply the row operation \( R_3 \leftarrow R_3 - R_1 \) to matrix \(A\). The third row of \(A\) is \([1, 2, 3]\) and the first row is \([1, 1, 1]\). Subtracting these gives \([1-1, 2-1, 3-1] = [0, 1, 2]\), which matches the third row of matrix \(B\).

Ensure All Other Rows Remain Unchanged

Check that no other rows are altered by this operation. Since only the third row is modified, rows one and two of matrices \(A\) and \(B\) remain consistent. This confirms that the operation \( R_3 \leftarrow R_3 - R_1 \) successfully transforms \(A\) into \(B\).

Key concepts

Elementary Row Operations

In the context of matrices, elementary row operations are powerful tools that enable us to manipulate matrices to solve systems of linear equations or simplify mathematical analysis. There are three primary types of elementary row operations:

• Row Switching: Interchanging two rows within the matrix. This does not alter the solution of a system of equations represented by the matrix.
• Row Multiplication: Multiplying all elements in a row by a non-zero scalar. This operation scales the row and is useful for making elements simpler or achieving row echelon form.
• Row Addition: Adding or subtracting a multiple of one row to another. This can be used to eliminate elements within a row, as demonstrated in our example where we transformed the first element of the third row from 1 to 0 by subtracting the first row from it: \( R_3 \leftarrow R_3 - R_1 \).

These operations are foundational for achieving row echelon form, simplifying matrices, and solving equations. When used properly, they maintain the equality represented by the matrix.

Matrix Transformation

Matrix transformation refers to the alteration of a matrix's appearance or structure through transformations such as rotations, scaling, or shearing operations. In linear algebra, one of the most practical transformations involves elementary row operations to simplify a matrix.

Through operations like multiplication or addition of rows, matrices can be taken from their initial cluttered state to simpler forms like row-echelon or reduced row-echelon forms. These forms reveal the solutions to systems of linear equations more clearly. For instance, transforming matrix \(A\) into \(B\) by \( R_3 \leftarrow R_3 - R_1 \) reveals a more straightforward pathway to understanding the system of equations represented by these matrices.

Transformations are not only theoretical but also practical. They serve in computer graphics when manipulating images or models, representing changes through matrices that directly impact the output image or structure.

System of Linear Equations

A system of linear equations is a set of equations in which each equation is linear in the variables. The end goal is to find the values of the variables that satisfy all the equations simultaneously. These systems can be expressed compactly using matrices.

• Each equation corresponds to a row in the matrix.
• Each variable is represented by a column.

For example, matrix \(A\) in our exercise can represent a system of linear equations, where each row corresponds to an equation.

Transforming the matrix using row operations can simplify these systems, making it straightforward to find solutions via techniques such as Gaussian elimination or matrix inversion. The transformation from \(A\) to \(B\) through the row operation illustrates how these equations can be adjusted to expose the solution more clearly.

The transformation ensures the system remains equivalent, meaning any solution sets derived from the matrix \(B\) are applicable to the original system described by matrix \(A\).