# What is the Determinant Function for a Matrix?

University of Oxford mathematician Dr Tom Crawford explains how to calculate the determinant of a 2×2 and a 3×3 matrix, as well as providing an insight into where the determinant function comes from. Check out ProPrep with a 30-day free trial to see how it can help you to improve your performance in STEM-based subjects – join here.

The video begins by presenting the definition of a 2×2 determinant for a general matrix as ad-bc. The concept of a determinant function mapping from matrices to scalars is then introduced, along with the three key properties that such a function must satisfy. These properties allow the uniqueness of determinants to be deduced. The 2×2 determinant formula is shown to satisfy the three required properties and therefore by appealing to uniqueness we can conclude it is in fact the only possible determinant for a 2×2 matrix. Next, the general formula for the determinant of a 3×3 matrix is introduced by expanding in the first row. The concept is then extended to other rows and columns of the matrix. Finally, a fully worked example of calculating the determinant of a 3×3 matrix is shown. First by expanding in row one, and then in row three where the zero entry helps to simplify the calculations.