Solve For Matrix B: A Step-by-Step Guide

by Pedro Alvarez 41 views

Hey guys! Today, we're diving deep into the fascinating world of matrices to solve a really cool problem. We've got a matrix equation, and our mission is to find the value of the unknown matrix B. This might sound intimidating, but trust me, we'll break it down into easy-to-understand steps. So, grab your thinking caps, and let's get started!

The Matrix Equation: Our Starting Point

First, let's lay out the problem. We're given matrix A:

A =  
  \begin{bmatrix}
    2 & 4 & -2 \\
    4 & -5 & 7 \\
    2 & 7 & 5
  \end{bmatrix}

And we know that when A is multiplied by another matrix B, we get the result:

AB =  
  \begin{bmatrix}
    24 \\
    -46 \\
    -2
  \end{bmatrix}

Our goal is to figure out what matrix B is. This is like solving a puzzle, where we need to find the missing piece that fits perfectly.

Understanding Matrix Multiplication

Before we jump into solving, let's quickly recap how matrix multiplication works. Remember, when we multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.

In our case, A is a 3x3 matrix (3 rows and 3 columns), and the result AB is a 3x1 matrix (3 rows and 1 column). This tells us that B must be a 3x1 matrix. Why? Because the number of columns in A (which is 3) must match the number of rows in B, and the result AB has 3 rows (from A) and 1 column (which will be the number of columns in B).

So, we can represent B as:

B =  
  \begin{bmatrix}
    x \\
    y \\
    z
  \end{bmatrix}

Where x, y, and z are the unknown values we need to find.

Setting Up the Equations

Now comes the fun part – setting up the equations! We'll use the rules of matrix multiplication to create a system of linear equations. Remember, to get the elements of the resulting matrix AB, we multiply the rows of A by the column of B.

Let's do it step-by-step:

  1. First row of A multiplied by B:

    (2 * x) + (4 * y) + (-2 * z) = 24

    This simplifies to:

    2x + 4y - 2z = 24

  2. Second row of A multiplied by B:

    (4 * x) + (-5 * y) + (7 * z) = -46

    This gives us:

    4x - 5y + 7z = -46

  3. Third row of A multiplied by B:

    (2 * x) + (7 * y) + (5 * z) = -2

    Which simplifies to:

    2x + 7y + 5z = -2

So, we now have a system of three linear equations:

2x + 4y - 2z = 24
4x - 5y + 7z = -46
2x + 7y + 5z = -2

Solving the System of Equations

Alright, we've got our equations ready. Now, we need to solve them to find the values of x, y, and z. There are several ways to do this, such as substitution, elimination, or using matrices (like Gaussian elimination or finding the inverse). For this example, let's use the elimination method, as it's often a straightforward approach.

Step 1: Simplify the First Equation

Notice that the first equation (2x + 4y - 2z = 24) has a common factor of 2. Let's divide the entire equation by 2 to simplify it:

x + 2y - z = 12

This makes our system of equations a bit easier to work with:

x + 2y - z = 12
4x - 5y + 7z = -46
2x + 7y + 5z = -2

Step 2: Eliminate x from the Second and Third Equations

To eliminate x, we'll use the first equation as our base. We'll multiply the first equation by -4 and add it to the second equation, and then multiply the first equation by -2 and add it to the third equation.

  • Eliminating x from the second equation:

    Multiply the first equation (x + 2y - z = 12) by -4:

    -4x - 8y + 4z = -48

    Add this to the second equation (4x - 5y + 7z = -46):

    (-4x - 8y + 4z) + (4x - 5y + 7z) = -48 + (-46)

    This simplifies to:

    -13y + 11z = -94

  • Eliminating x from the third equation:

    Multiply the first equation (x + 2y - z = 12) by -2:

    -2x - 4y + 2z = -24

    Add this to the third equation (2x + 7y + 5z = -2):

    (-2x - 4y + 2z) + (2x + 7y + 5z) = -24 + (-2)

    This simplifies to:

    3y + 7z = -26

Now our system of equations looks like this:

x + 2y - z = 12
-13y + 11z = -94
3y + 7z = -26

Step 3: Eliminate y from the Third Equation

Let's eliminate y from the third equation. To do this, we'll multiply the second equation (-13y + 11z = -94) by 3 and the third equation (3y + 7z = -26) by 13, and then add the two equations together.

  • Multiply the second equation by 3:

    3 * (-13y + 11z) = 3 * (-94)

    -39y + 33z = -282

  • Multiply the third equation by 13:

    13 * (3y + 7z) = 13 * (-26)

    39y + 91z = -338

  • Add the two equations:

    (-39y + 33z) + (39y + 91z) = -282 + (-338)

    This simplifies to:

    124z = -620

Step 4: Solve for z

Now we can easily solve for z:

z = -620 / 124 z = -5

Step 5: Solve for y

Substitute the value of z (-5) into the equation 3y + 7z = -26:

3y + 7 * (-5) = -26 3y - 35 = -26 3y = 9 y = 3

Step 6: Solve for x

Substitute the values of y (3) and z (-5) into the equation x + 2y - z = 12:

x + 2 * 3 - (-5) = 12 x + 6 + 5 = 12 x + 11 = 12 x = 1

The Solution: Matrix B Unveiled

We've done it! We've found the values of x, y, and z:

x = 1 y = 3 z = -5

Therefore, matrix B is:

B =  
  \begin{bmatrix}
    1 \\
    3 \\
    -5
  \end{bmatrix}

Checking Our Work

It's always a good idea to check our solution. Let's multiply matrix A by our calculated matrix B to see if we get the original result AB:

A =  
  \begin{bmatrix}
    2 & 4 & -2 \\
    4 & -5 & 7 \\
    2 & 7 & 5
  \end{bmatrix}

B =  
  \begin{bmatrix}
    1 \\
    3 \\
    -5
  \end{bmatrix}

AB =  
  \begin{bmatrix}
    (2*1) + (4*3) + (-2*-5) \\
    (4*1) + (-5*3) + (7*-5) \\
    (2*1) + (7*3) + (5*-5)
  \end{bmatrix}

AB =  
  \begin{bmatrix}
    2 + 12 + 10 \\
    4 - 15 - 35 \\
    2 + 21 - 25
  \end{bmatrix}

AB =  
  \begin{bmatrix}
    24 \\
    -46 \\
    -2
  \end{bmatrix}

Yes! Our result matches the given AB, so we know our solution for matrix B is correct.

Key Takeaways: Mastering Matrix Equations

Solving for an unknown matrix like B involves a few key steps:

  1. Understand Matrix Multiplication: Know the rules and how dimensions work.
  2. Set Up the Equations: Use matrix multiplication to create a system of linear equations.
  3. Solve the System: Use methods like elimination or substitution to find the unknown values.
  4. Check Your Work: Always verify your solution by plugging it back into the original equation.

By following these steps, you can confidently tackle matrix equations and unlock the hidden values within!

Conclusion: You're a Matrix Master!

Great job, guys! We successfully navigated through the world of matrices and solved for B. Remember, practice makes perfect, so keep exploring and tackling new matrix problems. You've got this! Matrix operations might seem daunting at first, but with a clear understanding of the underlying principles and a systematic approach, you can solve even the most complex problems. Keep practicing, and you'll become a true matrix master! Solving matrix equations is a fundamental skill in various fields, including computer graphics, engineering, and data science. Understanding how to find the unknown matrix in an equation is crucial for many applications. This step-by-step guide has equipped you with the tools and knowledge to confidently approach these types of problems. So, go forth and conquer those matrices!