Solving Matrix Equations: Finding The Value Of X

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 a matrix equation step-by-step and find the value of 'x' that makes it all work. Let's dive right in!

Understanding the Problem

Before we start crunching numbers, let's make sure we understand what we're dealing with. We're given the equation:

$\begin{pmatrix} 2 & x \\ -1 & 3 \\ \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 2 & 4 \\ \end{pmatrix} = \begin{pmatrix} 8 & 6 \\ 5 & 13 \\ \end{pmatrix}$

Our mission is to find the value of 'x' that satisfies this equation. In other words, we need to figure out what number 'x' needs to be so that when we multiply the two matrices on the left side, we get the matrix on the right side.

Matrix Multiplication Refresher

First things first, let's quickly recap how matrix multiplication works. When we multiply two matrices, we take the dot product of the rows of the first matrix with the columns of the second matrix. For example, if we have two matrices:

$A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$ and $B = \begin{pmatrix} e & f \\ g & h \\ \end{pmatrix}$

Then, their product $A \cdot B$ is:

$A \cdot B = \begin{pmatrix} a \cdot e + b \cdot g & a \cdot f + b \cdot h \\ c \cdot e + d \cdot g & c \cdot f + d \cdot h \\ \end{pmatrix}$

Each entry in the resulting matrix is the sum of the products of corresponding entries in the row of the first matrix and the column of the second matrix. Make sure you understand this process, because it's crucial for solving our problem.

Performing the Matrix Multiplication

Now that we've refreshed our memory on matrix multiplication, let's apply it to our equation. We need to multiply the two matrices on the left side:

$\begin{pmatrix} 2 & x \\ -1 & 3 \\ \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 2 & 4 \\ \end{pmatrix}$

Let's perform the multiplication step by step:

• Top-left entry: $(2 \cdot 1) + (x \cdot 2) = 2 + 2x$
• Top-right entry: $(2 \cdot -1) + (x \cdot 4) = -2 + 4x$
• Bottom-left entry: $(-1 \cdot 1) + (3 \cdot 2) = -1 + 6 = 5$
• Bottom-right entry: $(-1 \cdot -1) + (3 \cdot 4) = 1 + 12 = 13$

So, the resulting matrix from the multiplication is:

$\begin{pmatrix} 2 + 2x & -2 + 4x \\ 5 & 13 \\ \end{pmatrix}$

Setting Up the Equations

Now that we've performed the matrix multiplication, we can rewrite our original equation as:

$\begin{pmatrix} 2 + 2x & -2 + 4x \\ 5 & 13 \\ \end{pmatrix} = \begin{pmatrix} 8 & 6 \\ 5 & 13 \\ \end{pmatrix}$

For these two matrices to be equal, their corresponding entries must be equal. This gives us two equations:

1. $2 + 2x = 8$
2. $-2 + 4x = 6$

We can use either of these equations to solve for 'x'. Let's use the first equation, since it looks a bit simpler.

Solving for x

Let's solve the equation $2 + 2x = 8$ for 'x'.

1. Subtract 2 from both sides: $2x = 8 - 2$
2. Simplify: $2x = 6$
3. Divide both sides by 2: $x = \frac{6}{2}$
4. Simplify: $x = 3$

So, we've found that $x = 3$. You can also use the second equation $-2 + 4x = 6$ to verify that you get the same answer.

Verification

To be absolutely sure that our answer is correct, let's plug $x = 3$ back into the original matrix equation and see if it holds true.

$\begin{pmatrix} 2 & 3 \\ -1 & 3 \\ \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 2 & 4 \\ \end{pmatrix}$

Performing the multiplication again:

• Top-left entry: $(2 \cdot 1) + (3 \cdot 2) = 2 + 6 = 8$
• Top-right entry: $(2 \cdot -1) + (3 \cdot 4) = -2 + 12 = 10$
• Bottom-left entry: $(-1 \cdot 1) + (3 \cdot 2) = -1 + 6 = 5$
• Bottom-right entry: $(-1 \cdot -1) + (3 \cdot 4) = 1 + 12 = 13$

So, the resulting matrix is:

$\begin{pmatrix} 8 & 10 \\ 5 & 13 \\ \end{pmatrix}$

Wait a minute! The top-right entry doesn't match! We got 10 instead of 6. Let's recheck our calculations. Okay, it looks like I made a mistake on the top right entry $(2 \cdot -1) + (x \cdot 4) = -2 + 4x$. I should have verified this before. Let's go back and correct this.

Correcting Our Calculation

Let's solve the equation $-2 + 4x = 6$ for 'x'.

1. Add 2 to both sides: $4x = 6 + 2$
2. Simplify: $4x = 8$
3. Divide both sides by 4: $x = \frac{8}{4}$
4. Simplify: $x = 2$

Ah, I see! So, we've found that $x = 2$.

Verification (Again!)

Let's plug $x = 2$ back into the original matrix equation:

$\begin{pmatrix} 2 & 2 \\ -1 & 3 \\ \end{pmatrix} \begin{pmatrix} 1 & -1 \\ 2 & 4 \\ \end{pmatrix}$

Performing the multiplication again:

• Top-left entry: $(2 \cdot 1) + (2 \cdot 2) = 2 + 4 = 6$
• Top-right entry: $(2 \cdot -1) + (2 \cdot 4) = -2 + 8 = 6$
• Bottom-left entry: $(-1 \cdot 1) + (3 \cdot 2) = -1 + 6 = 5$
• Bottom-right entry: $(-1 \cdot -1) + (3 \cdot 4) = 1 + 12 = 13$

Woops! Top left should have been equal to 8. Let's try with the other equation.

Let's solve the equation $2 + 2x = 8$ for 'x'.

1. Subtract 2 from both sides: $2x = 8 - 2$
2. Simplify: $2x = 6$
3. Divide both sides by 2: $x = \frac{6}{2}$
4. Simplify: $x = 3$

Now, let's put x=3 in the top right formula. $-2 + 4(3) = -2 + 12 = 10$. Aha! There is an error in the original question! The top right corner can't be 6.

Conclusion

So, after working through the matrix multiplication and solving the equations, we found that x = 3 based on the top left entry. However, after plugging the x back in and checking the top right value, we see that there's an issue with the original matrix. In the original matrix it is not possible to make the top right entry equal to 6. Always double-check your work, and don't be afraid to go back and correct mistakes, guys! Matrix equations can be tricky, but with practice and a clear understanding of the steps involved, you'll become a pro in no time.