Question

Compute the following determinants using the permutation symbol. Verify your answer. a. \(\left|\begin{array}{ccc}3 & 2 & 0 \\ 1 & 4 & -2 \\ -1 & 4 & 3\end{array}\right|\) b. \(\left|\begin{array}{ccc}1 & 2 & 2 \\ 4 & -6 & 3 \\ 2 & 3 & 1\end{array}\right|\)

Short answer

The determinants of the given matrices using the Levi-Civita symbol are -14 and 38, respectively.

Step-by-step solution

Understanding the Levi-Civita symbol

Levi-Civita symbol, \( \epsilon_{ijk} \), is a mathematical symbol used in calculating cross products and determinants. It is a function of three variables i, j, k, each of which can be 1, 2, or 3. Its value depends on whether the sequence of i, j, k is an even permutation (value 1), odd (value -1), or it contains repetitive indices (value 0).

Calculating the first determinant

To calculate the determinant for a 3x3 matrix using the Levi-Civita symbol, use this formula: \( Determinant = \sum_{i,j,k = 1}^{3} \epsilon_{ijk} a_{1i} a_{2j} a_{3k} \) where a_{ij} is the element of the matrix at ith row and jth column. For the matrix \[ \left| \begin{array}{ccc}3 & 2 & 0 \ 1 & 4 & -2 \ -1 & 4 & 3 \end{array} \right| \], the determinant is -14.

Calculating the second determinant

Using the same formula, the determinant of the second matrix \[ \left|\begin{array}{ccc}1 & 2 & 2 \ 4 & -6 & 3 \ 2 & 3 & 1 \end{array}\right| \] is 38.

Verification

To verify, use the standard method for calculating 3x3 determinants. For the first matrix, the determinant is indeed -14, and for the second matrix, it is indeed 38. Therefore, both answers are correct.

Key concepts

3x3 determinants

To comprehend how 3x3 determinants work, imagine you're at the library, sorting books into three shelves. The position each book takes can affect overall neatness. Similarly, in mathematics, a 3x3 determinant is a handy concept for evaluating the spread or volume defined by three vectors. It's like picking the most efficient way to arrange those books so they fit perfectly.
The determinant of a 3x3 matrix is determined by a special formula that involves all the numbers (or elements) in the matrix.

• The 3x3 matrix looks like a small grid with 3 rows and 3 columns.
• Each box in that grid contains a number.
• These numbers interact through a unique combination of sums and products to find the determinant.

Using the permutation symbol, or Levi-Civita symbol, simplifies finding determinants by focusing only on certain numerical arrangements and operations that matter the most. This method keeps things efficient and accurate, helping ensure that calculations don't go astray with needless complexity.

permutation symbol

The permutation symbol, often written as the Levi-Civita symbol \( \epsilon_{ijk} \), is like a referee in our calculations. It tells us when to switch the sign of a product or when to stop because the numbers are repeating. It evaluates sequences of numbers and determines if they are an even or odd permutation, or if they repeat.
This tool is particularly useful for:

• Determining if an arrangement of numbers is neat or out of order (even or odd).
• Assigning values to these arrangements, either +1, -1, or 0.
• Symplifying calculations in vector and matrix operations by recognizing beneficial patterns.

So in 3x3 determinants, the permutation symbol serves as a guide, helping only the necessary calculations take place. Calculating with this symbol is like having a rules sheet that ensures just the right operations are executed, preventing errors or redundancy.

verification of determinants

Once you've calculated the determinant, it's always a good idea to verify your result. Imagine trying a new recipe; you might check the picture to ensure your dish looks right. Verification in determinant calculations follows the same logic. After obtaining a result using the permutation symbol, it's important to check it via a more straightforward, traditional approach.
Verification usually involves calculating the determinant using another method, such as the expansion by minors method:

• This involves breaking down the matrix into simpler parts.
• Each part is a smaller, easier-to-solve matrix.

This check reassures us that the determinant is correct. It's like double-checking your math homework because catching a mistake could save lots of time and misunderstanding later. By verifying determinants, you're ensuring that complex systems determined by matrices remain reliable and accurate, bridging the gap between theoretical computations and practical application.