Matrix Transpose And Equation: Finding X - Y

by TextBrain Team 45 views

In this comprehensive guide, we'll dive into a fascinating matrix problem involving matrix transposition and solving for unknown variables. Matrix operations are a fundamental concept in linear algebra and are widely used in various fields, including computer graphics, data analysis, and engineering. Let's break down this problem step by step so you can fully understand the concepts and techniques involved.

Understanding the Problem

The problem presents us with three matrices: A, B, and C. We are also given that Aᵀ represents the transpose of matrix A. The core of the problem lies in the equation 2Aᵀ + B = C. Our mission is to find the value of x - y, where x and y are elements within matrix B. This requires us to understand matrix transposition, scalar multiplication, and matrix addition. Let's first revisit these foundational concepts before tackling the problem head-on.

Matrix Transpose: Flipping Rows and Columns

The matrix transpose is a crucial operation. Think of it as flipping a matrix over its main diagonal (the diagonal from the top-left corner to the bottom-right corner). This effectively swaps the rows and columns. So, if matrix A is an m × n matrix (m rows and n columns), its transpose, Aᵀ, will be an n × m matrix. The element in the i-th row and j-th column of A becomes the element in the j-th row and i-th column of Aᵀ.

For example, if we have a matrix:

A = | 1  2 |
    | 3  4 |

Its transpose, Aᵀ, would be:

Aᵀ = | 1  3 |
     | 2  4 |

Scalar Multiplication: Scaling the Matrix

Scalar multiplication is another essential operation. It involves multiplying a matrix by a scalar (a single number). This is done by multiplying each element of the matrix by the scalar. If 'k' is a scalar and A is a matrix, then kA is obtained by multiplying every element in A by k.

For instance, if we have a matrix:

A = | 1  2 |
    | 3  4 |

And we multiply it by the scalar 2, we get:

2A = | 2  4 |
     | 6  8 |

Matrix Addition: Adding Corresponding Elements

Matrix addition is performed by adding the corresponding elements of two matrices. However, there's a crucial condition: the matrices must have the same dimensions (same number of rows and columns). If A and B are both m × n matrices, then their sum, A + B, is also an m × n matrix, where each element is the sum of the corresponding elements in A and B.

Consider two matrices:

A = | 1  2 |
    | 3  4 |

B = | 5  6 |
    | 7  8 |

Their sum, A + B, is:

A + B = | 6   8 |
        | 10 12 |

Now that we've refreshed our understanding of these fundamental matrix operations, let's get back to solving the problem.

Solving the Matrix Problem: A Step-by-Step Approach

Okay, guys, let's get into the nitty-gritty of solving this matrix problem. Remember, we're given:

A = | 3  0 |
    | 2  5 |

B = | x -1 |
    | y  1 |

C = | 0  3 |
    | 1 11 |

and the equation 2Aᵀ + B = C. Our goal is to find x - y.

Step 1: Find the Transpose of Matrix A (Aᵀ)

First things first, we need to find the transpose of matrix A. As we discussed, this involves swapping the rows and columns:

A = | 3  0 |
    | 2  5 |

So, the transpose Aᵀ is:

Aᵀ = | 3  2 |
     | 0  5 |

Step 2: Multiply Aᵀ by 2 (2Aᵀ)

Next, we perform scalar multiplication. We multiply each element of Aᵀ by 2:

Aᵀ = | 3  2 |
     | 0  5 |

2Aᵀ = | 2*3  2*2 |
      | 2*0  2*5 |

2Aᵀ = | 6  4 |
      | 0 10 |

Step 3: Set up the Equation 2Aᵀ + B = C

Now, we substitute 2Aᵀ, B, and C into the given equation:

2Aᵀ + B = C

| 6  4 |   | x -1 |   | 0  3 |
| 0 10 | + | y  1 | = | 1 11 |

Step 4: Perform Matrix Addition

We add the corresponding elements of 2Aᵀ and B:

| 6 + x   4 + (-1) |
| 0 + y  10 + 1    |

| 6 + x   3  |
| y     11 |

So, the equation becomes:

| 6 + x   3  |   | 0  3 |
| y     11 | = | 1 11 |

Step 5: Equate Corresponding Elements

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

  1. 6 + x = 0
  2. y = 1

Step 6: Solve for x and y

From the first equation, we solve for x:

6 + x = 0
x = -6

The second equation directly gives us:

y = 1

Step 7: Calculate x - y

Finally, we calculate x - y:

x - y = -6 - 1
x - y = -7

Conclusion: We Found Our Answer!

So, after carefully working through the steps, we've found that x - y = -7. This problem beautifully illustrates how matrix operations work together. Understanding matrix transposition, scalar multiplication, and matrix addition is crucial for solving such problems. Remember, practice makes perfect, so keep working on these types of problems!

By understanding the core principles and following a structured approach, you can confidently tackle any matrix problem that comes your way. Matrix algebra can seem daunting at first, but with practice, it becomes a powerful tool in your mathematical arsenal. This problem not only tests your knowledge of matrix operations but also your ability to apply those operations in a logical and sequential manner.

In summary, we successfully found the value of x - y by:

  1. Calculating the transpose of matrix A.
  2. Multiplying the transpose by a scalar (2).
  3. Setting up the equation 2Aᵀ + B = C.
  4. Performing matrix addition.
  5. Equating corresponding elements to form equations.
  6. Solving for x and y.
  7. Finally, calculating x - y.

I hope this step-by-step explanation has been helpful, guys. Keep practicing, and you'll master these matrix operations in no time! Remember, linear algebra is a powerful tool, and understanding it opens doors to many exciting fields and applications. Keep exploring, keep learning, and keep solving!