Ohio Assessments for Educators (OAE) Mathematics Practice Exam

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What formula represents the determinant of a 2x2 matrix?

ad + bc
ad - bc
ab + cd
ac - bd

The correct answer is: ad - bc

The determinant of a 2x2 matrix, which is generally represented as: \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] is calculated using the formula $ ad - bc $. In this context, $ a $ and $ d $ are the elements from the main diagonal of the matrix, while $ b $ and $ c $ are the elements from the secondary diagonal. This formula is derived from the method of cross multiplication, which gives the difference between the products of the diagonals. The first product $ ad $ comes from multiplying the diagonal elements going from the top left to the bottom right, while $ bc $ is the product from the top right to the bottom left. The significance of taking the difference is that it reflects how the area (or volume in higher dimensions) is affected by the linear transformation represented by the matrix. In the context of linear algebra, the determinant provides valuable insights, such as whether the matrix is invertible (a non-zero determinant indicates invertibility), and helps in finding the eigenvalues of the matrix among other applications.