Matrix Equation: Finding The Value Of C | Step-by-Step Solution

Hey guys! Today, we're diving into a fun matrix problem. We've got a matrix equation, $P = 2Q^t$, and our mission is to find the value of 'c'. Sounds like a plan? Let's break it down step by step so everyone can follow along. We'll make sure to keep things super clear and easy to understand. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we're all on the same page. We're given two matrices, P and Q, and a relationship between them: $P = 2Q^t$. This means matrix P is equal to 2 times the transpose of matrix Q. Our goal is to use this information to figure out the value of 'c'.

Here's what we know:

• $P = \begin{pmatrix} a & 4 \\ 2b & 3c \end{pmatrix}$
• $Q = \begin{pmatrix} 2c-3b & 2a+1 \\ a & b+7 \end{pmatrix}$

The superscript 't' in $Q^t$ means we need to find the transpose of matrix Q. Remember, the transpose of a matrix is found by swapping its rows and columns. This is a crucial step, so let’s make sure we get it right.

We will use the properties of matrix operations, such as scalar multiplication and transposition, to set up equations and solve for the unknowns. This problem combines several concepts from matrix algebra, so it’s a great way to test our understanding.

Step 1: Finding the Transpose of Matrix Q

The first thing we need to do is find the transpose of matrix Q, which we denote as $Q^t$. To find the transpose, we simply swap the rows and columns of Q. So, the first row of Q becomes the first column of $Q^t$, and the second row of Q becomes the second column of $Q^t$. Let's do it:

$Q = \begin{pmatrix} 2c-3b & 2a+1 \\ a & b+7 \end{pmatrix}$

Therefore, the transpose of Q is:

$Q^t = \begin{pmatrix} 2c-3b & a \\ 2a+1 & b+7 \end{pmatrix}$

Make sure you double-check this step, guys! Getting the transpose right is super important because it affects everything that comes after. Once we have $Q^t$, we can move on to the next part of the problem.

Now that we have the transpose, we can use it in the given equation to start solving for our unknowns. Remember, the equation we're working with is $P = 2Q^t$. We've found $Q^t$, so let's plug that into the equation and see what we get.

Step 2: Multiplying the Transpose of Q by 2

Now that we have $Q^t$, the next step is to multiply it by 2, as indicated in the equation $P = 2Q^t$. To do this, we simply multiply each element of the matrix $Q^t$ by 2. It’s like distributing the 2 across all the entries in the matrix. Here’s how it looks:

$2Q^t = 2 * \begin{pmatrix} 2c-3b & a \\ 2a+1 & b+7 \end{pmatrix} = \begin{pmatrix} 2(2c-3b) & 2a \\ 2(2a+1) & 2(b+7) \end{pmatrix} = \begin{pmatrix} 4c-6b & 2a \\ 4a+2 & 2b+14 \end{pmatrix}$

So, we've now found that $2Q^t = \begin{pmatrix} 4c-6b & 2a \\ 4a+2 & 2b+14 \end{pmatrix}$. This is a key step because we’re setting up the right side of our equation to match matrix P. We're getting closer to being able to equate the matrices and solve for our variables.

Take your time with scalar multiplication, guys. It's easy to make a small mistake, and that can throw off the whole solution. Double-check each entry to make sure you've multiplied correctly. Once we're confident in this result, we can move on to equating the matrices.

Step 3: Equating Matrices P and 2Qt

Okay, we're getting to the exciting part! Now that we have both P and $2Q^t$, we can equate them. Remember, $P = 2Q^t$. This means that each corresponding element in the two matrices must be equal. This gives us a set of equations that we can solve to find the values of a, b, and c.

Here are the matrices we're equating:

$P = \begin{pmatrix} a & 4 \\ 2b & 3c \end{pmatrix}$

$2Q^t = \begin{pmatrix} 4c-6b & 2a \\ 4a+2 & 2b+14 \end{pmatrix}$

So, equating the corresponding elements, we get the following equations:

1. $a = 4c - 6b$
2. $4 = 2a$
3. $2b = 4a + 2$
4. $3c = 2b + 14$

Now we have a system of equations! Our next step is to solve this system to find the values of a, b, and, most importantly, c. Don’t worry, we'll take it one equation at a time. Systems of equations can seem daunting, but if we break it down, it’s totally manageable.

Step 4: Solving the System of Equations

Alright, let's tackle this system of equations one by one. We have four equations:

1. $a = 4c - 6b$
2. $4 = 2a$
3. $2b = 4a + 2$
4. $3c = 2b + 14$

The easiest one to start with is equation (2), $4 = 2a$, because it only involves one variable. Let's solve for 'a':

$2a = 4$

$a = \frac{4}{2}$

$a = 2$

Great! We found that $a = 2$. Now we can substitute this value into other equations to solve for the remaining variables. Let's use equation (3), $2b = 4a + 2$, to find 'b':

$2b = 4(2) + 2$

$2b = 8 + 2$

$2b = 10$

$b = \frac{10}{2}$

$b = 5$

Awesome! We also found that $b = 5$. Now we can use these values to find 'c'. Let's use equation (4), $3c = 2b + 14$:

$3c = 2(5) + 14$

$3c = 10 + 14$

$3c = 24$

$c = \frac{24}{3}$

$c = 8$

We did it! We found that $c = 8$. We've successfully solved the system of equations. It's always satisfying when the pieces come together, right?

Step 5: Verifying the Solution (Optional but Recommended)

Okay, we’ve found a value for 'c', but it's always a good idea to double-check our work, guys. Math is like that – a little verification can save us from mistakes! We can plug the values we found (a = 2, b = 5, c = 8) back into the original equations to make sure they hold true. This will give us confidence that our solution is correct.

Let's quickly check the first equation, $a = 4c - 6b$:

$2 = 4(8) - 6(5)$

$2 = 32 - 30$

$2 = 2$ (This checks out!)

Now let’s check the fourth equation, $3c = 2b + 14$:

$3(8) = 2(5) + 14$

$24 = 10 + 14$

$24 = 24$ (This also checks out!)

Since our values satisfy the equations, we can be pretty confident that our solution is correct. Verifying the solution is a great habit to get into, especially on exams. It gives you that extra peace of mind.

Final Answer: The Value of C

Alright, after all that awesome work, we've arrived at our final answer! We successfully navigated through the matrix equation, solved the system of equations, and found the value of 'c'.

So, the value of c is 8.

Woohoo! Give yourselves a pat on the back, guys! You tackled a matrix problem like pros. Remember, the key is to break down the problem into smaller, manageable steps, and don’t be afraid to double-check your work.

I hope this step-by-step solution was clear and helpful. Matrix problems can seem tricky at first, but with practice and a good understanding of the concepts, you can conquer them all. Keep up the great work, and I'll see you in the next math adventure!