Matrix Multiplication: Finding X And Y When AB = C

by TextBrain Team 51 views

Hey guys! Today, we're diving into a super interesting matrix problem. We've got three matrices, A, B, and C, and we need to figure out the values of x and y that make the equation AB = C true. This involves some matrix multiplication and a little bit of algebra, so let's jump right in!

Setting Up the Problem

First, let's clearly define our matrices:

  • Matrix A:

    A=(35−xx1)A = \begin{pmatrix} 3 & 5-x \\ x & 1 \end{pmatrix}

  • Matrix B:

    B=(2y1x)B = \begin{pmatrix} 2 & y \\ 1 & x \end{pmatrix}

  • Matrix C:

    C=(152x−yy−2x−3y)C = \begin{pmatrix} 15 & 2x-y \\ y-2x & -3y \end{pmatrix}

Our mission, should we choose to accept it (and we do!), is to find the values of x and y that satisfy the equation AB = C. This means we need to multiply matrices A and B and then compare the result with matrix C. If these two values are equal, then the equation AB=C is true.

Understanding Matrix Multiplication

Before we start crunching numbers, let's quickly refresh how matrix multiplication works. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In our case, A and B are both 2x2 matrices, so we're good to go. The resulting matrix will also be a 2x2 matrix. Matrix multiplication is a fundamental operation in linear algebra, with vast applications in computer graphics, data analysis, and physics.

The element in the i-th row and j-th column of the product matrix is obtained by taking the dot product of the i-th row of the first matrix and the j-th column of the second matrix. Let's break this down:

  • (1,1) element of AB: (1st row of A) ⋅ (1st column of B)
  • (1,2) element of AB: (1st row of A) ⋅ (2nd column of B)
  • (2,1) element of AB: (2nd row of A) ⋅ (1st column of B)
  • (2,2) element of AB: (2nd row of A) ⋅ (2nd column of B)

Performing the Matrix Multiplication

Now, let's multiply matrices A and B:

AB=(35−xx1)(2y1x)AB = \begin{pmatrix} 3 & 5-x \\ x & 1 \end{pmatrix} \begin{pmatrix} 2 & y \\ 1 & x \end{pmatrix}

Let's calculate each element of the resulting matrix:

  • (1,1) element: (3 * 2) + ((5 - x) * 1) = 6 + 5 - x = 11 - x
  • (1,2) element: (3 * y) + ((5 - x) * x) = 3y + 5x - x^2
  • (2,1) element: (x * 2) + (1 * 1) = 2x + 1
  • (2,2) element: (x * y) + (1 * x) = xy + x

So, our resulting matrix AB is:

AB=(11−x3y+5x−x22x+1xy+x)AB = \begin{pmatrix} 11-x & 3y + 5x - x^2 \\ 2x+1 & xy+x \end{pmatrix}

Equating AB and C

We know that AB = C, so we can set the corresponding elements of the matrices equal to each other:

(11−x3y+5x−x22x+1xy+x)=(152x−yy−2x−3y)\begin{pmatrix} 11-x & 3y + 5x - x^2 \\ 2x+1 & xy+x \end{pmatrix} = \begin{pmatrix} 15 & 2x-y \\ y-2x & -3y \end{pmatrix}

This gives us a system of four equations:

  1. 11 - x = 15
  2. 3y + 5x - x^2 = 2x - y
  3. 2x + 1 = y - 2x
  4. xy + x = -3y

Now, it's time to solve this system of equations. Solving systems of equations is a crucial skill in various fields, including engineering, economics, and computer science. Let's tackle it step by step!

Solving the System of Equations

Let's start with the simplest equation, equation 1:

  1. 11 - x = 15 Solving for x, we get: x = 11 - 15 x = -4

Now that we have the value of x, we can substitute it into the other equations to solve for y. Let's use equation 3:

  1. 2x + 1 = y - 2x Substitute x = -4: 2(-4) + 1 = y - 2(-4) -8 + 1 = y + 8 -7 = y + 8 y = -15

So far, we have x = -4 and y = -15. To be sure, let's substitute these values into the remaining equations to check if they hold true. This is a crucial step to ensure our solutions are correct and consistent across the entire system. Verifying solutions is a common practice in mathematical problem-solving and helps prevent errors.

Verification

Let's verify the values in equations 2 and 4.

Equation 2:

  1. 3y + 5x - x^2 = 2x - y Substitute x = -4 and y = -15: 3(-15) + 5(-4) - (-4)^2 = 2(-4) - (-15) -45 - 20 - 16 = -8 + 15 -81 = 7

Oh no! This equation does not hold true. This indicates we might have made an error in our calculations or there might be no solution that satisfies all equations. Since equation 2 did not work, it's very important to go back and check our calculations. Let's carefully re-examine each step to find the mistake. This process highlights the importance of carefulness and verification in mathematical problem-solving.

Re-evaluating Equation 2

Let's go back to equation 2:

3y + 5x - x^2 = 2x - y

Plug in x = -4 and y = -15:

3(-15) + 5(-4) - (-4)^2 = 2(-4) - (-15) -45 - 20 - 16 = -8 + 15 -81 = 7

Okay, we still see the inconsistency. It seems there was no error in substitution; -81 does not equal 7. This strongly suggests there might not be a unique solution set that solves all equations simultaneously, or there was a mistake in formulating the original matrix equation.

Equation 4:

Let’s check Equation 4 just to confirm our suspicion that there’s an inconsistency:

  1. xy + x = -3y

Substitute x = -4 and y = -15:

(-4)(-15) + (-4) = -3(-15) 60 - 4 = 45 56 = 45

Again, this equation does not hold true. 56 does not equal 45. This further confirms the inconsistency within the system of equations derived from the matrix equation AB = C.

Conclusion on the System

Given our checks, there is a clear inconsistency: the values x = -4 and y = -15, which we derived from equations 1 and 3, do not satisfy equations 2 and 4. This could mean several things:

  1. There might be no solution to this system of equations.
  2. There might have been a mistake in the original problem statement or in setting up the equations from AB = C. For example, a typo in one of the matrix elements could lead to this inconsistency.

In real-world scenarios, encountering inconsistencies is a common part of mathematical modeling. It often indicates the need to re-evaluate the initial assumptions or the model setup.

Final Thoughts

This problem was a fantastic exercise in matrix multiplication and solving systems of equations. Although we didn't find a solution that satisfies all conditions, we learned a crucial lesson about verifying solutions and the possibility of inconsistencies in mathematical models. Remember, guys, it's not just about getting the answer; it's about understanding the process and recognizing potential issues along the way! Matrix operations like multiplication are fundamental in fields like computer science (especially in graphics), physics simulations, and data analysis. Next time, we might tweak the original matrices slightly and see if we can find consistent solutions, which is a typical approach when you suspect there might be a small error somewhere in your setup!

Keep practicing, and you'll become matrix multiplication masters in no time!