Question

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

Short answer

Multiply the second row of matrix A by 2 to obtain matrix B.

Step-by-step solution

Identify Changes from A to B

First, let's identify how matrix \(A\) must be modified to become matrix \(B\). Comparing the matrices, we observe that only the second row of matrix \(A\) changes to form matrix \(B\): It changes from \([1, 0, 1]\) to \([2, 0, 2]\).

Determine Row Operation Needed

Since we need the second row of \(A\) to become twice its original value to match the second row of \(B\), we apply a scalar multiplication row operation. Specifically, the transformation required is to multiply the entire second row of matrix \(A\) by 2.

Apply Row Operation

Perform the row operation on matrix \(A\): Multiply the second row by 2. This operation can be described as \(R_2 \rightarrow 2R_2\). Apply this operation to \(A\):Original second row: \([1, 0, 1]\).After operation: \([2 \times 1, 2 \times 0, 2 \times 1] = [2, 0, 2]\).

Verify the Result

After performing the row operation, verify the resulting matrix to ensure it matches matrix \(B\). The matrix \(A\) after the operation is:\[\begin{array}{ccc}1 & 1 & 1 \2 & 0 & 2 \1 & 2 & 3\end{array}\]This matches matrix \(B\) as required.

Key concepts

Row Operations in Matrices

Row operations in matrices are fundamental techniques utilized in matrix transformations, especially when converting one matrix into another through systematic adjustments. These operations are critical in solving linear systems or simplifying matrices to perform further computations. There are three primary types of row operations:

• Row Switching: This involves swapping two rows within the matrix. Such an operation changes the order but not the arithmetic content of the matrix.
• Row Multiplication: This operation multiplies a row by a non-zero scalar, changing the row's values while maintaining proportional relationships. For example, to change a row from \[ a, b, c \] to \[ ka, kb, kc \], you multiply every element by \(k\).
• Row Addition: This involves adding a scalar multiple of one row to another row. It's helpful in forming zero elements to simplify the matrix.

In the given exercise, the transformation of matrix \(A\) into matrix \(B\) makes use of the row multiplication operation, highlighting its utility in altering specific rows to match a desired matrix output. This particular example shows how a single operation can effectively transition a matrix's row into its required state.

Scalar Multiplication in Matrices

Scalar multiplication is a vital matrix operation and an integral part of transforming matrices in computations and linear algebra. It involves multiplying every element within a matrix by a constant, known as a scalar, effectively scaling the matrix without changing its structure.
Let's dive deeper into scalar multiplication within the context of the exercise. In the transformation of matrix \(A\) to matrix \(B\), only the second row is multiplied by a scalar, specifically 2, transforming \([1, 0, 1]\) into \([2, 0, 2]\).
This operation can be summarized as follows: if \[ R_i = [a, b, c] \], the transformation through scalar multiplication is \[ R_i \rightarrow kR_i \] which results in \[ [ka, kb, kc] \].

• It maintains the linearity of the row – all elements are scaled proportionally.
• The operation works independently of other rows, affecting only the selected row unless incorporated alongside other operations.

Scalar multiplication is instrumental in scenarios where specific rows need adjustment without disturbing the rest of the matrix, providing flexibility and precision in matrix manipulations.

Elementary Matrices

Elementary matrices represent a fundamental concept in linear algebra, instrumental in expressing and understanding row operations. An elementary matrix is formed by performing a single row operation on an identity matrix.
Their importance is underscored in various matrix computations such as finding inverses and simplifying larger matrices. By understanding elementary matrices, one can effectively track and replicate the row transformations in broader matrix operations.

• Row Multiplication Matrix: This type is formed by multiplying a row of an identity matrix by a nonzero scalar. It scales a specific row when multiplied with another matrix.
• Example: Consider the 3x3 identity matrix. Multiplying the second row by 2 forms a new elementary matrix that, when applied to another matrix, will perform \( R_2 \rightarrow 2R_2 \).
• Elementary matrices preserve matrix structure, enable reverse operations, and provide a way to visualize row operations.

In the context of the original exercise, understanding elementary matrices helps in conceptualizing how a simple row scaling operation can be represented systematically within matrix algebra, making complex transformations manageable and traceable.