Calculating The Determinant Coefficient Matrix For A System Of Equations

by Sharif Sakr 73 views

Hey everyone! Today, we're diving into the fascinating world of linear algebra to explore how to find the determinant of a coefficient matrix. Specifically, we'll be tackling the system of equations:

-x - y + 2z = 5
3x + 2y - z = 3
4x + 4y - 8z = -2

Our mission? To calculate the determinant of the coefficient matrix. So, grab your calculators (or your mental math muscles!) and let's get started!

Cracking the Code: Building the Coefficient Matrix

Before we can even think about determinants, we need to extract the coefficient matrix from our system of equations. Think of this matrix as the skeleton key that unlocks the secrets of the system. Each row in the matrix represents one equation, and each column corresponds to the coefficients of a specific variable (x, y, and z in this case). So, let's break it down:

  • Equation 1: -x - y + 2z = 5 gives us the coefficients -1, -1, and 2.
  • Equation 2: 3x + 2y - z = 3 gives us the coefficients 3, 2, and -1.
  • Equation 3: 4x + 4y - 8z = -2 gives us the coefficients 4, 4, and -8.

Now, we arrange these coefficients into a 3x3 matrix, which we'll call 'A':

A = | -1  -1   2 |
    |  3   2  -1 |
    |  4   4  -8 |

Ta-da! This is our coefficient matrix. It's a neat little package that holds the numerical relationships within our system of equations. Now comes the fun part: calculating the determinant!

The Determinant: A Deep Dive into Matrix Magic

The determinant is a special number associated with a square matrix (like our 3x3 matrix A). It's a scalar value, meaning it's just a single number, but it carries a ton of information about the matrix and the system of equations it represents. For instance, the determinant can tell us if the system has a unique solution, infinitely many solutions, or no solutions at all. It's like a secret code that reveals the nature of the system.

But how do we actually calculate this magical number? For a 3x3 matrix, we have a couple of popular methods. Let's explore one of the most common: the expansion by minors method.

Method 1: Expansion by Minors – Unveiling the Formula

The expansion by minors method might sound intimidating, but it's actually a systematic way of breaking down the 3x3 determinant into smaller, more manageable 2x2 determinants. Here's the general idea:

  1. Choose a row or column: You can pick any row or column to start with. For this example, let's stick with the first row (-1, -1, 2).
  2. Multiply each element by its cofactor: A cofactor is a signed minor. A minor is the determinant of the 2x2 matrix that remains after you cross out the row and column containing the element. The sign is determined by a checkerboard pattern of + and - signs, starting with + in the top-left corner.
  3. Sum the results: Add up the products you calculated in step 2, and you've got your determinant!

Let's apply this to our matrix A. We'll expand along the first row:

  • Element -1:
    • Minor: Cross out the first row and first column, leaving us with the 2x2 matrix | 2 -1 |. Its determinant is (2 * -8) - (-1 * 4) = -16 + 4 = -12. | 4 -8 |
    • Cofactor: Since -1 is in the top-left position (which has a + sign in our checkerboard), the cofactor is simply -12.
    • Product: (-1) * (-12) = 12
  • Element -1 (again!):
    • Minor: Cross out the first row and second column, leaving us with the 2x2 matrix | 3 -1 |. Its determinant is (3 * -8) - (-1 * 4) = -24 + 4 = -20. | 4 -8 |
    • Cofactor: Since -1 is in the first row, second column (which has a - sign in our checkerboard), the cofactor is -(-20) = 20.
    • Product: (-1) * 20 = -20
  • Element 2:
    • Minor: Cross out the first row and third column, leaving us with the 2x2 matrix | 3 2 |. Its determinant is (3 * 4) - (2 * 4) = 12 - 8 = 4. | 4 4 |
    • Cofactor: Since 2 is in the first row, third column (which has a + sign in our checkerboard), the cofactor is simply 4.
    • Product: 2 * 4 = 8

Now, we sum the products: 12 + (-20) + 8 = 0.

Therefore, the determinant of matrix A is 0!

Method 2: Sarrus's Rule – A Shortcut for 3x3 Matrices

If you're working specifically with 3x3 matrices, there's a handy shortcut called Sarrus's Rule. It's a visual method that can speed things up. Here's how it works:

  1. Rewrite the first two columns: Write the first two columns of the matrix again to the right of the original matrix.
  2. Calculate the diagonals:
    • Multiply the elements along the three diagonals going from top-left to bottom-right.
    • Multiply the elements along the three diagonals going from top-right to bottom-left.
  3. Sum and subtract: Add the products from the top-left to bottom-right diagonals. Subtract the products from the top-right to bottom-left diagonals. The result is the determinant!

Let's apply Sarrus's Rule to our matrix A:

| -1  -1   2 | -1  -1 |
|  3   2  -1 |  3   2 |
|  4   4  -8 |  4   4 |
  • Top-left to bottom-right diagonals:
    • (-1) * 2 * (-8) = 16
    • (-1) * (-1) * 4 = 4
    • 2 * 3 * 4 = 24
    • Sum: 16 + 4 + 24 = 44
  • Top-right to bottom-left diagonals:
    • 2 * 2 * 4 = 16
    • (-1) * (-1) * (-1) = -1
    • (-1) * 3 * (-8) = 24
    • Sum: 16 + (-1) + 24 = 39

Determinant = 44 - 39 = 5

Wait a minute! We got a different answer than with the expansion by minors method. This highlights a crucial point: Sarrus's Rule, while convenient, is only applicable to 3x3 matrices. It seems there was a calculation error in the previous method. Let's revisit the expansion by minors to pinpoint the mistake.

Backtracking and Correcting the Expansion by Minors

Okay, let's carefully re-examine the expansion by minors method. We'll stick to expanding along the first row:

  • Element -1:
    • Minor: As before, the determinant of | 2 -1 | is (2 * -8) - (-1 * 4) = -16 + 4 = -12. | 4 -8 |
    • Cofactor: Still -12.
    • Product: (-1) * (-12) = 12
  • Element -1 (again!):
    • Minor: The determinant of | 3 -1 | is (3 * -8) - (-1 * 4) = -24 + 4 = -20. This is correct. | 4 -8 |
    • Cofactor: The cofactor should be -(-20) = 20. This is also correct.
    • Product: (-1) * 20 = -20. This seems correct as well.
  • Element 2:
    • Minor: The determinant of | 3 2 | is (3 * 4) - (2 * 4) = 12 - 8 = 4. This is where the mistake was! | 4 4 |
    • Cofactor: The cofactor is 4.
    • Product: 2 * 4 = 8

Now, we sum the products: 12 + (-20) + 8 = 0.

So, even after careful recalculation, the expansion by minors method still yields a determinant of 0. This discrepancy between the two methods indicates a potential issue with the matrix itself. Let's investigate further.

The Zero Determinant: A Sign of Trouble (or Interesting Things!)

The fact that we're getting a determinant of 0 is a big clue. A zero determinant has significant implications for the system of equations:

  • No unique solution: A determinant of 0 means the system either has infinitely many solutions or no solutions at all. It indicates that the equations are not linearly independent, meaning one or more equations can be expressed as a combination of the others. Think of it like having redundant information; some equations aren't adding anything new to the mix.

  • Singular matrix: A matrix with a determinant of 0 is called a singular matrix. Singular matrices are non-invertible, meaning you can't find an inverse matrix that would