Solving Matrix Equations: A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem involving matrices. We're given a 2x2 matrix, cleverly named A, and some equations. Our goal is to figure out the result of a specific matrix multiplication. Don't worry; it sounds more complicated than it is! We'll break it down step by step to make it super clear. This problem is a great example of how we can use matrix algebra to solve systems of equations and understand linear transformations. Ready to get started? Let's go!

Understanding the Problem: Matrix Basics

First off, let's make sure we're all on the same page about what a matrix is. Think of a matrix like a grid of numbers arranged in rows and columns. In our case, matrix A is a 2x2 matrix, meaning it has two rows and two columns. We're also given two equations that involve matrix A:

1. A * [4 6]' = [0 2]'
2. A * [2 1]' = [0 1]'

Here, [4 6]' and [2 1]' are column vectors (matrices with only one column). The equations tell us how matrix A transforms these vectors. Our task is to find the result of A * [4 2; 2 3].

Now, let's decode what these equations mean. Each equation represents a linear transformation performed by matrix A. When A multiplies a vector, it essentially changes the vector's position or direction in space. The result is a new vector. The core concept we'll use here is the linearity of matrix multiplication: if you know how a matrix transforms certain vectors, you can figure out how it transforms a linear combination of those vectors. This principle is crucial in many areas of mathematics and physics, such as computer graphics and mechanics. Understanding this will make you a matrix master in no time!

Finding Matrix A: Step by Step

Alright, let's get down to business and find the actual values within matrix A. Since A is a 2x2 matrix, let's represent it with variables:

A = [[a, b], [c, d]]

Now, we'll use the equations given to us:

1. A * [4 6]' = [0 2]' which translates to: [[a, b], [c, d]] * [4, 6]' = [0, 2]'
2. A * [2 1]' = [0 1]' which translates to: [[a, b], [c, d]] * [2, 1]' = [0, 1]'

Performing the matrix multiplications in the first equation gives us:

• 4a + 6b = 0
• 4c + 6d = 2

And from the second equation, we get:

• 2a + 1b = 0
• 2c + 1d = 1

Now we have a system of equations! We can solve these to find the values of a, b, c, and d. Let's rearrange the equations a bit to make them easier to solve. From 2a + b = 0, we can say b = -2a. Substituting this into 4a + 6b = 0, we get 4a + 6(-2a) = 0 which simplifies to -8a = 0. Therefore, a = 0. Since b = -2a, then b = 0 as well.

Similarly, from 2c + d = 1, we can write d = 1 - 2c. Substitute this into 4c + 6d = 2, which gives us 4c + 6(1 - 2c) = 2. This simplifies to -8c + 6 = 2, or -8c = -4, meaning c = 1/2. Finally, because d = 1 - 2c, then d = 1 - 2(1/2) = 0.

So, our matrix A is:

A = [[0, 0], [1/2, 0]]

See? It wasn’t too bad, right? We've just found the components of the matrix that satisfies the given conditions. The trick is to break it down into simpler steps and use the properties of matrices to our advantage. This will definitely make you feel like a math wizard!

Calculating the Final Result: Putting it All Together

Now that we know what matrix A is, we can find the result of A * [[4, 2], [2, 3]]. Let's do the multiplication:

[[0, 0], [1/2, 0]] * [[4, 2], [2, 3]]

When performing the matrix multiplication, remember that you multiply each row of the first matrix by each column of the second matrix and sum the results. Doing this, we get:

• First row: (0 * 4) + (0 * 2) = 0, (0 * 2) + (0 * 3) = 0
• Second row: (1/2 * 4) + (0 * 2) = 2, (1/2 * 2) + (0 * 3) = 1

So the result of the multiplication is:

[[0, 0], [2, 1]]

And there you have it! We've successfully calculated the final result. It’s all about the steps and keeping track of what you're doing. We began with understanding the basic matrix operations and applied the equations to discover matrix A, then applied it to the target matrix multiplication to obtain our final result. The key is to always remember the properties of matrices and break down each step into manageable parts.

Conclusion: Matrix Mastery Achieved

We did it, guys! We started with a matrix equation, found the unknown matrix A, and then used it to solve a new matrix multiplication. This exercise shows us how useful matrices can be in solving complex problems. We used the given information, broke down the matrix components, and solved a system of equations. This not only helped us in understanding how matrices work, but also gave us a practical example of how we can use them. The process involved understanding the transformation a matrix provides and applying basic matrix operations to find our answer. By following each step, we could get the correct solution. Great job, everyone! You are all officially matrix masters. Keep practicing, and you'll become even more confident in solving matrix problems! And remember, the more you practice, the easier it will get!