Ohio Assessments for Educators (OAE) Mathematics Practice Exam

When calculating a determinant of a larger matrix, what is the first step?

1. Find the product of all elements.

2. Take the average of all elements.

3. Take the determinant of the smaller matrix.

4. Transpose the matrix.

The correct answer is: Take the determinant of the smaller matrix.

The first step in calculating the determinant of a larger matrix often involves simplifying the calculation process by breaking it down into smaller components. Hence, calculating the determinant of a smaller matrix is essential, as one can use methods like expansion by minors or cofactor expansion. This process entails selecting a row or a column from the larger matrix and expressing the determinant in terms of the determinants of smaller matrices created from that selection. By taking determinants of these smaller matrices recursively, you eventually reduce the problem to computing determinants of 2x2 or 3x3 matrices, which can be done directly using their specific formulas. Finding the product of all elements, taking the average, or transposing the matrix are not effective or relevant steps in the determinant calculation process. Each of these alternatives fails to contribute to deriving the determinant, which requires systematic reduction or manipulation of the matrix rather than mere operations on its elements.