Prime Number Matrix & Solving Matrix Equations: Math Problems

Hey guys! Let's dive into some matrix problems today. We're going to tackle creating a matrix with prime numbers and then solve a matrix equation. Buckle up, it's going to be a mathematical adventure!

1. Creating a Prime Number Matrix

Okay, so the first part asks us to create a matrix. Specifically, we need a matrix that has 4 rows and 3 columns. The catch? All the elements (the numbers inside the matrix) must be prime numbers. But what even are prime numbers? Well, prime numbers are whole numbers greater than 1 that have only two divisors: 1 and themselves. Think 2, 3, 5, 7, 11, and so on. They're the cool kids of the number world because they can't be divided evenly by anyone except 1 and themselves.

To build this prime number matrix, we'll systematically fill in the slots. We'll need 4 rows and 3 columns, so that's 4 x 3 = 12 prime numbers in total. Let's start with the smallest prime numbers and work our way up. This is where understanding the definition of prime numbers really comes in handy, guys. We need to make sure each number we put in is only divisible by 1 and itself. This means skipping numbers like 4 (divisible by 2), 6 (divisible by 2 and 3), 8 (divisible by 2 and 4), and so on. We're looking for those pure, indivisible primes!

Let's break it down step-by-step. The first prime number is 2, a classic! Then comes 3, another solid choice. Next up is 5, followed by 7. See how we're skipping numbers like 4 and 6? Those aren't prime. The next few prime numbers are 11, 13, 17, 19, 23, 29, 31, and 37. That gives us 12 prime numbers to fill our matrix. Once we've identified these prime numbers, the next step is arranging them in the matrix format – 4 rows and 3 columns. We can arrange these numbers in various ways, and each arrangement will result in a valid matrix that satisfies the given conditions. Remember, the order matters in a matrix, so even though the elements are the same, a different arrangement creates a different matrix. This is what makes matrix creation so interesting – there's often more than one right answer!

Now, let’s consider what type of matrix we’ve created. A matrix is simply a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Since our matrix has a specific number of rows and columns (4 rows and 3 columns), it's a rectangular matrix. There isn't a more specific classification we can give it based solely on the fact that its elements are prime numbers. If the matrix had the same number of rows and columns, it would be a square matrix. If it had only one row, it would be a row matrix, and so on. But in our case, it's simply a rectangular matrix filled with prime numbers. This is important because the shape of the matrix can affect the types of operations we can perform on it, such as matrix multiplication.

Here's an example of what the matrix might look like:

$\begin{pmatrix} 2 & 3 & 5 \ 7 & 11 & 13 \ 17 & 19 & 23 \ 29 & 31 & 37 \end{pmatrix}$

So, there you have it! We've successfully created a 4x3 matrix where every element is a prime number, and we've identified it as a rectangular matrix. Pat yourself on the back – you've conquered the first matrix challenge!

2. Solving a Matrix Equation

Alright, let's move on to the second part of the problem: solving a matrix equation. This might sound intimidating, but don't worry, we'll break it down step-by-step, guys. The equation we're given looks like this:

$\begin{pmatrix} P-9 & Q+r \ 3S+r & 2P-4S \end{pmatrix} = \begin{pmatrix} 16 & 2 \ 14 & 12 \end{pmatrix}$

What this equation is telling us is that these two matrices are equal. And what does it mean for two matrices to be equal? It means that their corresponding elements must be equal. In other words, the element in the first row and first column of the first matrix must be equal to the element in the first row and first column of the second matrix, and so on for all the elements. This is the key to solving this problem, guys. We're going to use this principle to create a system of equations and then solve for our unknowns: P, Q, r, and S.

Let's start by equating the corresponding elements. We get the following equations:

• P - 9 = 16
• Q + r = 2
• 3S + r = 14
• 2P - 4S = 12

Now we have a system of four equations with four unknowns. This is a classic algebra problem! There are several ways we can solve this system, such as substitution, elimination, or using matrix methods (ironically!). Let's use substitution and elimination, since those are the most straightforward for this particular problem. The first equation, P - 9 = 16, is the easiest to solve for P. We simply add 9 to both sides to get P = 25. See? We've already found one of our variables! That's a great start, guys!

Now that we know P, we can substitute it into the fourth equation, 2P - 4S = 12, to solve for S. Substituting P = 25, we get 2(25) - 4S = 12, which simplifies to 50 - 4S = 12. Subtracting 50 from both sides gives us -4S = -38, and dividing both sides by -4 gives us S = 9.5. So, we've found S as well! We're on a roll, guys!

Next, we have two equations involving Q and r: Q + r = 2 and 3S + r = 14. We already know S = 9.5, so we can substitute that into the second equation to solve for r. We get 3(9.5) + r = 14, which simplifies to 28.5 + r = 14. Subtracting 28.5 from both sides gives us r = -14.5. Now we have r! Just one variable left to find, guys!

Finally, we can substitute the value of r into the equation Q + r = 2 to solve for Q. We get Q + (-14.5) = 2, which simplifies to Q - 14.5 = 2. Adding 14.5 to both sides gives us Q = 16.5. And there we have it! We've solved for all the variables: P = 25, S = 9.5, r = -14.5, and Q = 16.5. We've conquered the matrix equation!

In summary, here's what we found:

• P = 25
• Q = 16.5
• r = -14.5
• S = 9.5

Conclusion

So, we've tackled two different types of matrix problems today. We created a matrix filled with prime numbers and classified it as a rectangular matrix, and we solved a matrix equation by setting up and solving a system of equations. These are fundamental concepts in linear algebra, and understanding them is crucial for further studies in mathematics, physics, computer science, and many other fields. Great job, guys! You're becoming matrix masters!