Lower Triangular Matrix: Finding The Value Of 3a^2 - 2bc - 4

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Hey guys! Let's dive into the fascinating world of matrices, specifically lower triangular matrices. We've got a cool problem to solve today that involves finding the value of a mathematical expression given a lower triangular matrix. This isn't just about crunching numbers; it's about understanding the properties of matrices and how they work. So, grab your thinking caps, and let's get started!

What's a Lower Triangular Matrix Anyway?

Before we jump into the problem, let's quickly recap what a lower triangular matrix is. In simple terms, a lower triangular matrix is a square matrix where all the elements above the main diagonal are zero. The main diagonal runs from the top-left corner to the bottom-right corner of the matrix. Think of it like a triangle of zeros sitting on top of the matrix. This unique structure gives lower triangular matrices some special properties, which we'll use to solve our problem.

To illustrate, consider a general 3x3 matrix:

[[a, b, c],
 [d, e, f],
 [g, h, i]]

For this matrix to be lower triangular, the elements b, c, and f must be zero. So, a lower triangular matrix looks something like this:

[[a, 0, 0],
 [d, e, 0],
 [g, h, i]]

See how all the elements above the main diagonal (0, 0, and 0) are zero? That's the key characteristic of a lower triangular matrix. This property is super important because it gives us a set of equations that we can use to solve for unknowns within the matrix. In our case, we'll use this property to find the values of a, b, and c, which we then need to calculate the final expression.

The Problem at Hand

Here's the matrix we're working with:

P = [[2, -3, 1],
     [3a-b, 5, -1],
     [c+2, 2b-6, 5]]

We're told that this matrix P is a lower triangular matrix. Our mission, should we choose to accept it (and we do!), is to find the value of the expression 3a² - 2bc - 4. Sounds like a plan, right? Let's break it down step by step. The problem combines matrix properties with algebraic manipulation, making it a great exercise in mathematical thinking. Remember, the goal isn't just to get the right answer but also to understand the process of solving the problem. This understanding will help you tackle similar problems in the future with confidence.

Cracking the Matrix Code: Setting Up the Equations

Since P is a lower triangular matrix, we know that all the elements above the main diagonal must be zero. This gives us three crucial equations:

  1. -3 = 0
  2. 1 = 0
  3. -1 = 0

Wait a minute! Something's not right. These equations seem contradictory. -3 cannot equal 0, 1 cannot equal 0, and -1 definitely cannot equal 0. This indicates there's an error in the matrix provided in the problem. The matrix P cannot be a lower triangular matrix with the values given. To proceed, we need to correct the matrix. Let's assume the matrix P is actually:

P = [[2, 0, 0],
     [3a-b, 5, 0],
     [c+2, 2b-6, 5]]

Now, the elements above the main diagonal are indeed zero, which aligns with the definition of a lower triangular matrix. This correction is crucial because it allows us to set up the correct equations and solve for the unknowns.

With the corrected matrix, we can now focus on the elements that were originally non-zero above the diagonal. These elements must be equal to zero for the matrix to be lower triangular. This gives us the following equations:

  1. 3a - b = 0
  2. 2b - 6 = 0
  3. c + 2 = 0

These equations are the key to unlocking the values of a, b, and c. Each equation provides a relationship between the variables, and by solving them, we can determine the individual values. This is where our algebra skills come into play. We'll use techniques like substitution and elimination to isolate the variables and find their values. Remember, the goal is to find the values of a, b, and c so that we can plug them into the expression 3a² - 2bc - 4 and get our final answer.

Solving for a, b, and c: A Step-by-Step Guide

Let's tackle these equations one by one. The second equation, 2b - 6 = 0, looks like the easiest place to start. We can isolate b with a couple of simple steps:

  1. Add 6 to both sides: 2b = 6
  2. Divide both sides by 2: b = 3

Boom! We've found the value of b. Now that we know b = 3, we can use this information to solve for a in the first equation, 3a - b = 0:

  1. Substitute b = 3: 3a - 3 = 0
  2. Add 3 to both sides: 3a = 3
  3. Divide both sides by 3: a = 1

Awesome! We've got a = 1 and b = 3. Only c left to go. Let's use the third equation, c + 2 = 0:

  1. Subtract 2 from both sides: c = -2

Fantastic! We've successfully decoded the matrix and found all the values: a = 1, b = 3, and c = -2. This was like cracking a secret code, wasn't it? Now that we have these values, we're ready for the final step: plugging them into the expression 3a² - 2bc - 4.

The Grand Finale: Plugging in the Values

Now comes the moment we've been working towards! We have a = 1, b = 3, c = -2, and our expression is 3a² - 2bc - 4. Let's substitute these values into the expression:

3(1)² - 2(3)(-2) - 4

Now, let's simplify step by step, following the order of operations (PEMDAS/BODMAS):

  1. Calculate the exponent: 3(1) - 2(3)(-2) - 4
  2. Multiply: 3 + 12 - 4
  3. Add and subtract from left to right: 15 - 4 = 11

Ta-da! The value of the expression 3a² - 2bc - 4 is 11.

Wrapping It Up

So, there you have it! We successfully navigated the world of lower triangular matrices, set up equations, solved for unknowns, and finally, calculated the value of the expression 3a² - 2bc - 4. This problem was a great example of how different mathematical concepts – matrices, algebra, and order of operations – come together to solve a single problem.

Remember, the key to mastering math isn't just memorizing formulas but understanding the underlying concepts and how they connect. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!