Problem 66 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The row operation \(k R_{i}+R_{j}\) indicates that it is the elements in row \(i\) that change.

Step-by-Step Solution

Answer

The statement is false. The correct statement should be: The row operation \(k R_{i}+R_{j}\) indicates that it is the elements in row \(j\) that change.

Step 1: Understand the operation

The given operation of the form \(k R_{i} + R_{j}\) represents a row operation on a matrix where \(k\) is a constant and \(R_{i}\) and \(R_{j}\) are rows in the matrix. It indicates that we should replace the \(R_{j}\) row with the result of the multiplication of row \(i\) by a scalar \(k\) plus the \(R_{j}\) row.

Step 2: Determine if the statement is correct

The statement claims that operation \(k R_{i}+R_{j}\) changes elements in row \(i\). Analyzing the operation, we see that we indeed manipulate values of row \(i\) by multiplying by \(k\), but we then add it to row \(j\) and replace row \(j\) with the result. The elements in row \(i\) remain unchanged, so this statement is False.

Step 3: Correct the false statement

Since the statement is false, we must adjust it to reflect the correct operation, which states that elements in row \(j\) are the ones that change when the operation \(k R_{i}+R_{j}\) is performed, not elements in row \(i\).