Matrix Inverse: Find X + Y Value | FGV-SP Question

Hey guys! Let's dive into an interesting matrix problem today. We're tackling a question from FGV-SP that involves matrix inverses. This is a fundamental concept in linear algebra, and understanding it is crucial for various applications in mathematics, physics, computer science, and engineering. So, let's break it down step by step and make sure we nail it!

Understanding the Problem

The question states that matrix A is the inverse of matrix B. This is a key piece of information! Remember, if two matrices are inverses of each other, their product results in the identity matrix. The identity matrix, often denoted as I, is a square matrix with ones on the main diagonal and zeros everywhere else. For a 2x2 matrix, the identity matrix looks like this:

I = [[1, 0],
     [0, 1]]

So, mathematically, if A is the inverse of B, then:

A * B = I

and

B * A = I

In our specific problem, we have:

A = [[x, 1],
     [5, 3]]

and

B = [[3, -1],
     [y, 2]]

Our mission, should we choose to accept it (and we do!), is to find the value of x + y. We can achieve this by utilizing the property of inverse matrices we just discussed. We'll multiply matrix A by matrix B, set the result equal to the identity matrix, and then solve for x and y. Sounds like a plan, right?

The Matrix Multiplication

Let's perform the matrix multiplication A * B:

[[x, 1],   *  [[3, -1],
 [5, 3]]      [y, 2]]

Remember the rules of matrix multiplication? We multiply the rows of the first matrix by the columns of the second matrix. So, the resulting matrix will be:

[[(x*3 + 1*y), (x*(-1) + 1*2)],
 [(5*3 + 3*y), (5*(-1) + 3*2)]]

Simplifying this, we get:

[[3x + y, -x + 2],
 [15 + 3y, 1]]

Setting Up the Equations

Now, we know that A * B must equal the identity matrix:

[[3x + y, -x + 2],
 [15 + 3y, 1]] = [[1, 0],
                 [0, 1]]

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

1. 3x + y = 1
2. -x + 2 = 0
3. 15 + 3y = 0
4. 1 = 1 (This equation doesn't give us any new information, but it confirms our calculations are on the right track)

Solving for x and y

We have a system of equations, and now it's time to put on our problem-solving hats and find the values of x and y. Let's start with the second equation, as it looks the simplest:

-x + 2 = 0

Adding x to both sides, we get:

2 = x

So, x = 2! That was easy, right? Now, let's move on to the third equation:

15 + 3y = 0

Subtracting 15 from both sides, we have:

3y = -15

Dividing both sides by 3, we find:

y = -5

Great! We've found that y = -5. Just to be sure, let's plug these values of x and y into the first equation to check if they hold true:

3x + y = 1
3(2) + (-5) = 1
6 - 5 = 1
1 = 1

Yep, it checks out! So, we've confidently determined that x = 2 and y = -5.

Finding x + y

The final step is to calculate the sum of x and y:

x + y = 2 + (-5) = -3

Wait a minute! -3 isn't one of the answer choices (A) 1 (B) 2 (C) 3 (D) 4. Let's double-check our work. Did we make a mistake somewhere? It's always a good idea to review your steps, especially in math problems.

Okay, going back through our calculations, we see a small oversight! When looking at the matrix multiplication, specifically the bottom left element, we have 15 + 3y = 0. We correctly solved this to get y = -5. However, let's look at the top left element: 3x + y = 1. We found x = 2 from the equation -x + 2 = 0. Plugging these values into 3x + y = 1 gives us 3(2) + (-5) = 1, which simplifies to 6 - 5 = 1, and that's correct. So where did we go wrong?

Ah, the crucial step we missed is recognizing that we need both A * B = I and B * A = I to confirm that A and B are indeed inverses. We only calculated A * B. Let's calculate B * A:

B * A = [[3, -1],   *  [[x,  1],
         [y,  2]]      [5,  3]]

Performing the multiplication:

[[(3*x + -1*5), (3*1 + -1*3)],
 [(y*x + 2*5), (y*1 + 2*3)]]

Simplifying:

[[3x - 5, 0],
 [xy + 10, y + 6]]

Now, we set this equal to the identity matrix:

[[3x - 5, 0],
 [xy + 10, y + 6]] = [[1, 0],
                     [0, 1]]

This gives us the following equations:

1. 3x - 5 = 1
2. 0 = 0 (This is good, it confirms part of our calculation)
3. xy + 10 = 0
4. y + 6 = 1

Let's solve these equations. From equation 1:

3x - 5 = 1
3x = 6
x = 2

This confirms our earlier result for x. From equation 4:

y + 6 = 1
y = -5

This also confirms our earlier result for y. Now, let's use equation 3 to double-check. If our values for x and y are correct, they should satisfy xy + 10 = 0:

(2)*(-5) + 10 = 0
-10 + 10 = 0
0 = 0

Okay, it all checks out! So, we have x = 2 and y = -5. The error wasn't in our calculations, but in the fact that I initially made a mistake in presenting the final answer options analysis in the previous attempt (apologies for that!).

Now, as before, we calculate the sum:

x + y = 2 + (-5) = -3

But we remember the answer options were (A) 1 (B) 2 (C) 3 (D) 4, so there's still something amiss.

Let's refocus on the equations we derived from B * A = I , especially the equation y + 6 = 1, which gave us y = -5. Also, 3x - 5 = 1 which correctly gave us x = 2. We ALSO have the equation xy + 10 = 0. Plugging in our values, we got (2)(-5) + 10 = 0, which is true.

From A * B = I, we had -x + 2 = 0, leading to x = 2, and 15 + 3y = 0, leading to y = -5. The other crucial equation was 3x + y = 1. Substituting gives 3(2) - 5 = 1, so 6 - 5 = 1, which is correct.

I pinpointed my mistake! It's in the original analysis. Let's backtrack and solve this carefully one more time. The correct calculation is x + y = 2 + (-5) = -3. However, looking back at the original FGV – SP question and its multiple-choice answers, the options provided are (A) 1 (B) 2 (C) 3 (D) 4. Since -3 does not appear as one of the options, this suggests there might be an error either in the problem statement itself or in the provided answer choices. Given our thorough calculations and verifications, it's highly likely that there's an issue with the question as presented. If this were an actual exam, I'd double-check my work as we've done here, and if confident in the result, I'd either mark the closest option or, if permitted, note the discrepancy to the examiner.

Key Takeaways

• The product of two inverse matrices is the identity matrix.
• Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix.
• Setting up a system of equations is crucial for solving matrix problems.
• Always double-check your work, especially in math problems!
• Don't just calculate A * B = I; you MUST also verify B * A = I to confirm inverses.
• If your answer doesn't match the choices, carefully review each step, but also consider the possibility of an error in the question itself.

I hope this breakdown was helpful, guys! Matrix problems can seem daunting at first, but with a systematic approach and careful attention to detail, you can conquer them. Keep practicing, and you'll become a matrix master in no time!