Final answer:
The question is about calculating the determinant using various methods: expanding along rows, expanding along columns, using Laplace's expansion, and applying the inverse matrix. Each method is explained step-by-step, with formulas for finding the determinant using its corresponding cofactors or inverse matrix.
Step-by-step explanation:
The given question is about calculating the determinant using various methods. The options provided are:
- Expand along rows
- Expand along columns
- Use Laplace's expansion
- Apply the inverse matrix
The determinant can be calculated using any of these methods, depending on the given matrix. Let's take each option one by one and understand how it is done.
A) Expand along rows:
To find the determinant by expanding along rows, you can use the formula:
Determinant = (a × Cofactor of a) + (b × Cofactor of b) + (c × Cofactor of c)
Where a, b, and c are the coefficients of the first row, and Cofactor of a, Cofactor of b, and Cofactor of c are the respective cofactors. You can calculate the determinant by repeating this process for each row.
B) Expand along columns:
Expanding along columns is similar to expanding along rows, but instead, you consider the coefficients of the first column and their respective cofactors. The formula would be:
Determinant = (a × Cofactor of a) + (d × Cofactor of d) + (g × Cofactor of g)
You would repeat this process for each column.
C) Use Laplace's expansion:
Laplace's expansion method involves expanding the determinant along a row or column using cofactors determined from the remaining elements. You can choose any row or column and use the formula:
Determinant = a × Cofactor of a - b × Cofactor of b + c × Cofactor of c
The sign alternates in each term, starting with positive.
D) Apply the inverse matrix:
If you have the inverse matrix available, you can easily calculate the determinant using the formula:
Determinant = 1 / (a × a-1) + 1 / (b × b-1) + 1 / (c × c-1)
Where a-1, b-1, and c-1 are the respective elements of the inverse matrix..