Finding The Inverse Of Matrix C (C = A - B)

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Hey guys! Let's dive into a cool math problem involving matrices. We're given two matrices, A and B, and we need to find the inverse of matrix C, where C is the result of subtracting matrix B from matrix A. Sounds like fun, right? Let's break it down step by step.

Understanding Matrices A, B, and C

First, let's define our players. We have matrix A:

A = egin{pmatrix} 3 & 4 \ 2 & 3 egin{pmatrix}

And matrix B:

B = egin{pmatrix} 4 & 5 \ -3 & -4 egin{pmatrix}

Our mission, should we choose to accept it (and we do!), is to find the inverse of matrix C. But what is C? Well, C is simply A minus B. So, before we can find the inverse, we need to figure out what C actually is. This involves subtracting corresponding elements of matrices B from A. Remember, matrix subtraction (and addition) is element-wise. This means we subtract the element in the first row and first column of B from the element in the first row and first column of A, and so on for all the elements.

Let’s perform the subtraction:

C = A - B = egin{pmatrix} 3 & 4 \ 2 & 3 egin{pmatrix} - egin{pmatrix} 4 & 5 \ -3 & -4 egin{pmatrix} = egin{pmatrix} 3-4 & 4-5 \ 2-(-3) & 3-(-4) egin{pmatrix}

Performing these subtractions, we get:

C = egin{pmatrix} -1 & -1 \ 5 & 7 egin{pmatrix}

Now that we have matrix C, we are one step closer to our goal. The next big step is to actually find the inverse of this resulting matrix. But first, it’s crucial to understand what a matrix inverse is and how we calculate it. Stay with me, we're getting there!

Calculating Matrix C

Okay, so we've established that finding matrix C is the first crucial step. We do this by subtracting matrix B from matrix A, element by element. Think of it like subtracting two grids – you subtract the numbers in the same positions from each other. This might seem simple, but it's a fundamental operation in linear algebra, and getting it right is essential for everything that follows.

Let's reiterate the matrices we're working with:

A = egin{pmatrix} 3 & 4 \ 2 & 3 egin{pmatrix}
B = egin{pmatrix} 4 & 5 \ -3 & -4 egin{pmatrix}

Now, let's perform the subtraction:

  • Top-left element: 3 - 4 = -1
  • Top-right element: 4 - 5 = -1
  • Bottom-left element: 2 - (-3) = 2 + 3 = 5
  • Bottom-right element: 3 - (-4) = 3 + 4 = 7

Putting these results together, we get matrix C:

C = egin{pmatrix} -1 & -1 \ 5 & 7 egin{pmatrix}

Fantastic! We've successfully calculated matrix C. This is a significant milestone because we can't find the inverse of C without knowing what C is! Now that we have C, our next task is to figure out how to find its inverse. This is where things get a little more interesting, involving concepts like determinants and adjugate matrices. But don't worry, we'll take it one step at a time.

Finding the Determinant of Matrix C

Alright, now that we've got our matrix C, the next key step in finding its inverse is to calculate the determinant of C. The determinant is a special number that can be calculated from a square matrix, and it tells us a lot about the matrix, including whether it even has an inverse. If the determinant is zero, the matrix is singular and doesn't have an inverse. So, finding the determinant is like checking if the door to the inverse even exists!

Remember our matrix C?

C = egin{pmatrix} -1 & -1 \ 5 & 7 egin{pmatrix}

For a 2x2 matrix like this, the determinant is calculated with a simple formula:

det(C) = (a*d) - (b*c)

Where a, b, c, and d are the elements of the matrix:

C = egin{pmatrix} a & b \ c & d egin{pmatrix}

In our case:

  • a = -1
  • b = -1
  • c = 5
  • d = 7

So, let's plug those values into our formula:

det(C) = (-1 * 7) - (-1 * 5) = -7 - (-5) = -7 + 5 = -2

Great! The determinant of matrix C is -2. Since the determinant is not zero, this is excellent news! It means that matrix C is invertible, and we can proceed to find its inverse. The non-zero determinant is like a green light, telling us we're on the right track. Now that we have the determinant, we're one step closer to unlocking the inverse of C. Our next move involves finding the adjugate (or adjoint) of matrix C.

Calculating the Adjugate (Adjoint) of Matrix C

Okay, so we've successfully calculated the determinant of matrix C, which is -2. That's awesome! Now, we need to find something called the adjugate (sometimes called the adjoint) of matrix C. The adjugate is like a modified version of our original matrix, and it's a crucial component in calculating the inverse.

Remember our matrix C?

C = egin{pmatrix} -1 & -1 \ 5 & 7 egin{pmatrix}

For a 2x2 matrix, finding the adjugate is a pretty straightforward process. We do two things:

  1. Swap the positions of the elements on the main diagonal (the diagonal from the top-left to the bottom-right).
  2. Change the signs of the elements on the off-diagonal (the diagonal from the top-right to the bottom-left).

Let's apply these steps to matrix C:

  1. Swap the main diagonal elements: The -1 and 7 swap places.
  2. Change the signs of the off-diagonal elements: The -1 and 5 become 1 and -5, respectively.

This gives us the adjugate of C, which we'll denote as adj(C):

adj(C) = egin{pmatrix} 7 & 1 \ -5 & -1 egin{pmatrix}

See? It's not too scary! The adjugate is essentially a rearrangement and sign-flipping operation. With the adjugate in hand, we're in the home stretch. We have the determinant, we have the adjugate, and now we're ready to put it all together to find the inverse of matrix C. Let's move on to the final step!

Determining the Inverse of Matrix C

Alright, guys! We've reached the final stage! We've calculated the determinant of matrix C (which was -2), and we've found the adjugate of matrix C. Now, the moment we've been waiting for: finding the inverse of matrix C! This is where all our hard work pays off.

The inverse of a matrix, which we'll denote as C⁻¹, is calculated using the following formula:

C⁻¹ = (1 / det(C)) * adj(C)

In simpler terms, we divide the adjugate of C by the determinant of C. We already know both of these pieces:

  • det(C) = -2
  • adj(C) = egin{pmatrix} 7 & 1 \ -5 & -1 egin{pmatrix}

So, let's plug them into the formula:

C⁻¹ = (1 / -2) * egin{pmatrix} 7 & 1 \ -5 & -1 egin{pmatrix}

This means we need to multiply every element in the adjugate matrix by -1/2:

C⁻¹ = egin{pmatrix} (1/-2)*7 & (1/-2)*1 \ (1/-2)*-5 & (1/-2)*-1 egin{pmatrix}

Performing the multiplication, we get:

C⁻¹ = egin{pmatrix} -7/2 & -1/2 \ 5/2 & 1/2 egin{pmatrix}

And there we have it! We've successfully calculated the inverse of matrix C. It might look a little messy with those fractions, but that's perfectly normal. The inverse matrix C⁻¹ is:

C⁻¹ = egin{pmatrix} -7/2 & -1/2 \ 5/2 & 1/2 egin{pmatrix}

Conclusion

So, there you have it! We've walked through the entire process of finding the inverse of a matrix, starting from subtracting matrices to calculating determinants and adjugates. We successfully found that the inverse of matrix C is:

C⁻¹ = egin{pmatrix} -7/2 & -1/2 \ 5/2 & 1/2 egin{pmatrix}

Hopefully, this breakdown has made the process clear and less intimidating. Matrix inverses might seem complex at first, but with practice and a step-by-step approach, you can totally master them! Keep practicing, and you'll be solving matrix problems like a pro in no time!