Solving The Matrix Equation: A Step-by-Step Guide

Hey guys! Ever stumbled upon a matrix equation and felt a bit lost? Don't worry, it happens to the best of us. Today, we're going to break down how to solve a matrix equation, specifically the one you've presented: $\begin{bmatrix} 6 & 4 \\ 9 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$. We'll go through each step in detail, so you'll be a matrix-solving pro in no time! Let's dive in and make math a little less intimidating and a lot more fun.

Understanding Matrix Equations

Before we jump into solving, let's make sure we're all on the same page about what a matrix equation actually is. Think of it as a compact way to represent a system of linear equations. In our case, the matrix equation $\begin{bmatrix} 6 & 4 \\ 9 & 6 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$ is just a shorthand for the following two equations:

• 6x + 4y = 1
• 9x + 6y = 3

See? It's not so scary when you break it down. The matrix on the left ($\begin{bmatrix} 6 & 4 \\ 9 & 6 \end{bmatrix}$) contains the coefficients of our variables (x and y). The column vector ($\begin{bmatrix} x \\ y \end{bmatrix}$) represents the variables themselves, and the column vector on the right ($\begin{bmatrix} 1 \\ 3 \end{bmatrix}$) contains the constants. Understanding this connection between matrix equations and systems of linear equations is key to solving them.

So, why use matrices in the first place? Well, matrices provide a really organized and efficient way to handle systems of equations, especially when dealing with more variables and equations. They also open the door to some powerful techniques for solving these systems, which we'll explore in a bit. Plus, matrix notation is super common in various fields like computer graphics, physics, and engineering, so getting comfortable with it is a smart move. For example, in computer graphics, matrices are used to perform transformations like rotations, scaling, and translations of objects in 3D space. In physics, they can represent things like the moment of inertia of a rigid body or the relationships between forces and displacements in a mechanical system. And in engineering, matrices are used extensively in structural analysis, circuit design, and control systems. So, the skills you're building here are definitely transferable to a wide range of applications!

Methods to Solve Matrix Equations

Alright, now that we know what we're dealing with, let's talk about the different ways we can actually solve a matrix equation like the one we have. There are a few main approaches, each with its own strengths and weaknesses. We'll focus on a couple of the most common and effective methods:

1. Gaussian Elimination (Row Reduction): This method involves systematically manipulating the rows of the augmented matrix (which we'll explain in a moment) to transform it into a simpler form, called row-echelon form or reduced row-echelon form. From this simplified form, we can easily read off the solutions for our variables. Gaussian elimination is a powerful technique that works for any system of linear equations, regardless of the number of equations or variables. It's a bit like a systematic puzzle-solving strategy. We're going to use row operations – things like adding multiples of one row to another or swapping rows – to simplify the matrix. The goal is to get the matrix into a form where it's easy to see the solutions for x and y.

2. Inverse Matrix Method: If the coefficient matrix (the one with the 6, 4, 9, and 6 in our case) has an inverse, we can use it to solve the equation directly. The inverse of a matrix, denoted by A⁻¹, is like the reciprocal of a number in regular algebra. When you multiply a matrix by its inverse, you get the identity matrix (a matrix with 1s on the diagonal and 0s everywhere else). This method is particularly useful when you need to solve the same matrix equation with different constant vectors (the $\begin{bmatrix} 1 \\ 3 \end{bmatrix}$ part). Once you've found the inverse, you can just multiply it by the new constant vector to get the solution. However, this method only works if the matrix has an inverse, which isn't always the case. If the determinant of the matrix is zero, then the inverse doesn't exist, and we need to use a different method.

We'll be diving into the Gaussian elimination method in detail for this particular problem. While the inverse matrix method is cool, it can be a bit more computationally intensive for 2x2 matrices, and Gaussian elimination gives us a nice, straightforward approach.

Solving with Gaussian Elimination: Step-by-Step

Okay, let's get our hands dirty and solve this matrix equation using Gaussian elimination! Remember, the goal is to transform the matrix into a form where we can easily read off the values of x and y.

Step 1: Form the Augmented Matrix

The first thing we need to do is create what's called the augmented matrix. This is simply the coefficient matrix ($\begin{bmatrix} 6 & 4 \\ 9 & 6 \end{bmatrix}$) with the constant vector ($\begin{bmatrix} 1 \\ 3 \end{bmatrix}$) tacked onto the end as an extra column. We usually draw a vertical line to separate the coefficient matrix from the constant vector. So, our augmented matrix looks like this:

$\begin{bmatrix} 6 & 4 & | & 1 \\ 9 & 6 & | & 3 \end{bmatrix}$

This augmented matrix represents our system of equations in a compact form. The first two columns correspond to the coefficients of x and y, and the third column represents the constants on the right-hand side of the equations.

Step 2: Perform Row Operations to Get to Row-Echelon Form

Now comes the fun part: manipulating the rows of the matrix! We're allowed to perform three basic row operations:

1. Swap two rows: This is like changing the order of the equations in our system.
2. Multiply a row by a non-zero constant: This is like multiplying both sides of an equation by the same number.
3. Add a multiple of one row to another row: This is like adding a multiple of one equation to another.

Our goal is to use these operations to get the matrix into row-echelon form. This means we want to get a leading 1 (called a pivot) in the first row, first column, and a 0 below it. Then, we want a leading 1 in the second row, second column. Let's see how it works in practice.

• First, let's get a leading 1 in the first row, first column. We can do this by dividing the first row by 6:

  $\begin{bmatrix} 6 & 4 & | & 1 \\ 9 & 6 & | & 3 \end{bmatrix}$ -> $\begin{bmatrix} 1 & 2/3 & | & 1/6 \\ 9 & 6 & | & 3 \end{bmatrix}$ (Dividing Row 1 by 6)

• Next, we want to get a 0 below that leading 1. We can do this by subtracting 9 times the first row from the second row:

  $\begin{bmatrix} 1 & 2/3 & | & 1/6 \\ 9 & 6 & | & 3 \end{bmatrix}$ -> $\begin{bmatrix} 1 & 2/3 & | & 1/6 \\ 0 & 0 & | & 3/2 \end{bmatrix}$ (Subtracting 9 * Row 1 from Row 2)

Whoa, hold on a second! Look at that second row. We've got 0s in the first two columns, and then 3/2 in the last column. This translates back to the equation 0x + 0y = 3/2, which is clearly impossible! This means our system of equations is inconsistent, and there's no solution. This can happen sometimes, and it's important to recognize it. This indicates that the two original equations represent parallel lines that never intersect. Therefore, there is no pair of x and y values that can satisfy both equations simultaneously.

Why No Solution?

The fact that we ended up with 0 = 3/2 tells us something important about our original equations. Remember, we started with:

• 6x + 4y = 1
• 9x + 6y = 3

Notice anything special about these equations? If we multiply the first equation by 3/2, we get:

(3/2) * (6x + 4y) = (3/2) * 1

9x + 6y = 3/2

Now compare this to the second equation: 9x + 6y = 3. We have the same left-hand side (9x + 6y), but different right-hand sides (3/2 and 3). This means there's no way for both equations to be true at the same time. They represent parallel lines, as mentioned before, and parallel lines never intersect. This is why our Gaussian elimination process led us to an impossible equation.

Key Takeaways

So, what have we learned today, guys? We've covered a lot about solving matrix equations, including:

• How matrix equations represent systems of linear equations.
• The Gaussian elimination method for solving matrix equations.
• How to form the augmented matrix.
• How to perform row operations to simplify the matrix.
• How to recognize an inconsistent system (one with no solution).

And most importantly, we've seen that sometimes, there's just no solution, and that's okay! It's all part of the matrix-solving adventure. The key is to understand the methods, apply them carefully, and interpret the results correctly. Whether you're dealing with math problems, engineering challenges, or computer graphics, the ability to work with matrices is a valuable skill to have. Keep practicing, and you'll become a matrix master in no time!