How To Calculate The Determinant Of A 4x4 Matrix

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Calculating the determinant of a 4x4 matrix might seem daunting at first, but with a systematic approach, it becomes a manageable task. In this comprehensive guide, we will explore the process of finding the determinant of the given 4x4 matrix:

| 0  1  0  1 |
| 0  2  0  0 |
| 0  3  1  0 |
| 1  4  0  0 |

We'll break down the steps, explain the underlying principles, and provide clear explanations to help you master this essential linear algebra concept. Understanding determinants is crucial in various fields, including solving systems of linear equations, finding eigenvalues, and determining the invertibility of a matrix.

Understanding Determinants

Before diving into the calculation, let's briefly discuss what a determinant is. The determinant of a square matrix is a scalar value that can be computed from the elements of the matrix. It provides valuable information about the matrix, such as whether the matrix has an inverse (a non-zero determinant indicates invertibility) and the volume scaling factor of a linear transformation represented by the matrix. For a 2x2 matrix, the determinant is calculated as follows:

| a  b |
| c  d | = ad - bc

However, for larger matrices like our 4x4 matrix, the calculation is more involved. We'll employ a technique called cofactor expansion to simplify the process. The main keywords here are: determinant, 4x4 matrix, cofactor expansion, and linear algebra. These keywords will help in understanding the concept and the methods used for calculation.

Method 1: Cofactor Expansion Along the First Column

The most efficient way to find the determinant of this matrix is by using cofactor expansion. Cofactor expansion allows us to break down a larger determinant into smaller, more manageable determinants. We can choose any row or column to perform the expansion, but selecting a row or column with the most zeros can simplify the calculation. In our case, the first column has three zeros, making it an ideal choice. The general formula for cofactor expansion along the first column is:

det(A) = a₁₁C₁₁ + a₂₁C₂₁ + a₃₁C₃₁ + a₄₁C₄₁

Where aᡒ₁ represents the elements in the first column, and Cᡒ₁ represents the corresponding cofactors. A cofactor is the determinant of the submatrix formed by removing the i-th row and first column, multiplied by (-1)^(i+1). Applying this to our matrix:

det = 0 * C₁₁ + 0 * C₂₁ + 0 * C₃₁ + 1 * C₄₁

Notice how the zeros eliminate the need to calculate the first three cofactors, significantly simplifying the problem. We only need to calculate C₄₁. The cofactor C₄₁ is found by removing the fourth row and first column from the original matrix, and then finding the determinant of the resulting 3x3 matrix:

C₄₁ = (-1)^(4+1) * det | 1  0  1 |
                    | 2  0  0 |
                    | 3  1  0 |

The term (-1)^(4+1) simplifies to -1, so we have:

C₄₁ = -1 * det | 1  0  1 |
            | 2  0  0 |
            | 3  1  0 |

Now, we need to find the determinant of this 3x3 matrix. We can again use cofactor expansion, this time choosing the second column as it contains two zeros. Expanding along the second column, we get:

det | 1  0  1 | = 0 * (...) + 0 * (...) + 1 * (-1)^(3+2) * det | 1  1 |
    | 2  0  0 |                                            | 2  0 |
    | 3  1  0 |

The determinant of the 2x2 matrix | 1 1 | is (1 * 0) - (1 * 2) = -2. So, | 2 0 |

det | 1  0  1 | = 1 * (-1) * (-2) = 2
    | 2  0  0 |
    | 3  1  0 |

Substituting this back into our equation for C₄₁:

C₄₁ = -1 * 2 = -2

Finally, substituting C₄₁ back into the original determinant equation:

det = 1 * C₄₁ = 1 * (-2) = -2

Therefore, the determinant of the given 4x4 matrix is -2. The cofactor expansion method simplifies the determinant calculation, especially when dealing with matrices containing zeros. Understanding how to choose the optimal row or column for expansion is crucial for efficient calculation. The main keywords in this section are: cofactor expansion, 3x3 determinant, submatrix, and determinant calculation. These terms highlight the specific steps and techniques used in this method.

Method 2: Step-by-Step Calculation of the 3x3 Determinant

Let's delve deeper into the calculation of the 3x3 determinant we encountered in the previous method. This will provide a more granular understanding of how 3x3 determinants are computed, which is a fundamental skill in linear algebra. We are calculating the determinant of the following matrix:

| 1  0  1 |
| 2  0  0 |
| 3  1  0 |

As we previously determined, expanding along the second column is the most efficient approach due to the two zeros. However, let's also explore how to calculate the determinant by expanding along the first row to illustrate the process further. The formula for cofactor expansion along the first row is:

det(A) = a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃

Where a₁ᡒ represents the elements in the first row, and C₁ᡒ represents the corresponding cofactors. Applying this to our 3x3 matrix:

det = 1 * C₁₁ + 0 * C₁₂ + 1 * C₁₃

The cofactors are calculated as follows:

C₁₁ = (-1)^(1+1) * det | 0  0 | = 1 * (0*0 - 0*1) = 0
                    | 1  0 |

C₁₂ = (-1)^(1+2) * det | 2  0 | = -1 * (2*0 - 0*3) = 0
                    | 3  0 |

C₁₃ = (-1)^(1+3) * det | 2  0 | = 1 * (2*1 - 0*3) = 2
                    | 3  1 |

Substituting these values back into the determinant equation:

det = 1 * 0 + 0 * 0 + 1 * 2 = 2

This confirms our previous result obtained through expansion along the second column. The step-by-step calculation of the 3x3 determinant involves finding the cofactors of each element and then applying the cofactor expansion formula. The main keywords in this section are: 3x3 determinant, cofactor, step-by-step calculation, and linear algebra. These keywords emphasize the detailed process of computing the determinant.

Method 3: General Formula for 3x3 Determinant Calculation

Another way to calculate the determinant of a 3x3 matrix is by using a direct formula. This formula is derived from the cofactor expansion method but provides a more streamlined approach for calculation. Given a 3x3 matrix:

| a  b  c |
| d  e  f |
| g  h  i |

The determinant can be calculated using the following formula:

det = a(ei - fh) - b(di - fg) + c(dh - eg)

Let's apply this formula to our 3x3 matrix:

| 1  0  1 |
| 2  0  0 |
| 3  1  0 |

Here, a = 1, b = 0, c = 1, d = 2, e = 0, f = 0, g = 3, h = 1, and i = 0. Substituting these values into the formula:

det = 1(0*0 - 0*1) - 0(2*0 - 0*3) + 1(2*1 - 0*3)
    = 1(0) - 0(0) + 1(2)
    = 0 - 0 + 2
    = 2

Again, we arrive at the same result, which is 2. The general formula provides a direct and efficient method for calculating the determinant of a 3x3 matrix. However, it's essential to understand the underlying principles of cofactor expansion to appreciate how this formula is derived. The main keywords in this section are: general formula, 3x3 matrix, determinant, and direct calculation. These keywords highlight the efficiency and directness of this method.

Conclusion

In conclusion, we have successfully calculated the determinant of the given 4x4 matrix, which is -2. We achieved this by employing the cofactor expansion method, which is a powerful technique for simplifying determinant calculations. We also explored the detailed calculation of 3x3 determinants using both cofactor expansion and the general formula. Understanding determinants is fundamental in linear algebra and has applications in various fields. Mastering these calculation techniques will significantly enhance your problem-solving abilities in mathematics and related disciplines. The main keywords throughout this guide are: determinant, 4x4 matrix, cofactor expansion, 3x3 determinant, linear algebra, and determinant calculation. These keywords are essential for grasping the core concepts and methods discussed in this article. By understanding these concepts, you can tackle more complex matrix problems with confidence.

Therefore, the correct answer is:

C. -2