Matrix Transformation Explained Transforming Matrix I To Matrix II
Hey guys! Today, we're diving deep into a linear algebra problem that involves transforming matrices. Specifically, we're going to break down which operation properly transforms matrix I into matrix II. This is a fundamental concept in linear algebra, and mastering it will help you tackle more complex problems with confidence. So, let's jump right in and explore the step-by-step process of matrix transformation. We will explore how row operations can be used to simplify matrices and ultimately solve systems of linear equations. Buckle up, and let's get started!
Understanding Matrix Transformations
Before we get into the specifics of this problem, let's quickly recap what matrix transformations are all about. In essence, matrix transformations involve manipulating the rows of a matrix using elementary row operations. These operations are crucial for solving systems of linear equations and simplifying matrices into more manageable forms. Think of it like this: we're trying to rearrange the information within the matrix in a way that makes it easier to read and interpret. The main goal is often to get the matrix into row-echelon form or reduced row-echelon form, which makes it straightforward to find solutions.
Elementary Row Operations: The Key to Transformation
The heart of matrix transformations lies in three elementary row operations:
- Swapping two rows: This is like rearranging the order of equations in a system. For example, if you have a matrix where the second row has a leading 1 but the first row doesn't, swapping them can be a great first step.
- Multiplying a row by a non-zero constant: This is equivalent to multiplying both sides of an equation by a constant. It's incredibly useful for creating leading 1s or eliminating fractions.
- Adding a multiple of one row to another: This operation is akin to adding multiples of equations together in a system. It’s a workhorse for eliminating variables and simplifying the matrix.
These operations are the tools we'll use to navigate from Matrix I to Matrix II. They might seem simple, but their power lies in how we combine them strategically.
The Goal: Simplifying and Solving
The ultimate aim of these transformations is usually to simplify the matrix so that it represents a system of equations that's easy to solve. When a matrix is in row-echelon form or reduced row-echelon form, the solution becomes apparent. In these forms, we can easily read off the values of the variables or back-substitute to find them. Understanding this goal helps us choose the right operations to perform. We're not just randomly manipulating rows; we're doing it with a purpose—to reveal the solution.
Analyzing Matrix I and Matrix II
Now, let's take a closer look at the matrices we're working with. We have Matrix I:
[ 1 2 1 | 5 ]
[ 0 1 2 | 5 ]
[ 2 7 8 | 25 ]
and Matrix II:
[ 1 2 1 | 5 ]
[ 0 1 2 | 5 ]
[ 0 3 6 | 15 ]
Our mission is to figure out what operation was performed to transform Matrix I into Matrix II. To do this, we’ll compare the two matrices row by row and see what changes have occurred. This is like detective work – we're looking for clues to uncover the transformation.
Spotting the Differences
The first two rows of both matrices are identical, which is a good start. This tells us that whatever operation was performed, it didn’t affect the first two rows. This observation narrows down our search significantly. We can focus our attention on the third row, where the change must have happened.
Looking at the third row, we see that in Matrix I, it's [2 7 8 | 25], and in Matrix II, it's [0 3 6 | 15]. The change is quite noticeable, especially the first element going from 2 to 0. This suggests that we're dealing with the third elementary row operation: adding a multiple of one row to another. The key is to figure out which row and what multiple were used.
Identifying the Operation
To transform the first element of the third row from 2 to 0, we likely used the first row. The first row has a 1 in the first position, so if we multiply the first row by -2 and add it to the third row, we’ll get the desired 0. Let's test this out. This process of identifying the operation often involves some trial and error, but with practice, you'll develop an intuition for which operations are most likely.
Performing the Row Operation
Let's perform the row operation we've identified. We're going to multiply the first row of Matrix I by -2 and add the result to the third row. Here’s how it looks:
-2 * Row 1: [-2 -4 -2 | -10]
Adding this to Row 3 of Matrix I: [2 7 8 | 25]
Result: [0 3 6 | 15]
Lo and behold, this matches the third row of Matrix II! This confirms that the operation we identified is indeed the one used to transform Matrix I into Matrix II. It’s always satisfying when the pieces of the puzzle come together, isn't it?
Writing it Out Formally
In mathematical notation, we can express this operation as:
R3 -> R3 - 2R1
This notation is a concise way of saying