Oxford Linear Algebra: What is the Determinant Function for a Matrix?

University of Oxford mathematician Dr Tom Crawford explains how to calculate the determinant of a 2x2 and a 3x3 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: https://www.proprep.uk/info/TOMROCKSMATHS **Sorry for the audio quality my microphone battery died :( Test your understanding with some practice exercises courtesy of ProPrep. You can download the workbooks and solutions for free at the links below. Determinants: https://www.proprep.uk/Academic/DownloadBook?file=Proprep%20-%20Linear%20Algebra%20-%20Determinanat%20-%20workbook%20uk.pdf You can also find fully worked video solutions from ProPrep instructors at the links below. 2x2 matrices: https://www.proprep.uk/general-modules/all/linear-algebra/determinants/determinants/vid9881/ 3x3 matrices: https://www.proprep.uk/general-modules/all/linear-algebra/determinants/determinants/vid9882/ Watch other videos from the Oxford Linear Algebra series at the links below. Solving Systems of Linear Equations using Elementary Row Operations (ERO’s): https://youtu.be/9pF__coVyEE Calculating the inverse of 2x2, 3x3 and 4x4 matrices: https://youtu.be/VKOaG3Ogf9Q The video begins by presenting the definition of a 2x2 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 2x2 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 2x2 matrix. Next, the general formula for the determinant of a 3x3 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 3x3 matrix is shown. First by expanding in row one, and then in row three where the zero entry helps to simplify the calculations. Produced by Dr Tom Crawford at the University of Oxford. Tom is an Early-Career Teaching and Outreach Fellow at St Edmund Hall: https://www.seh.ox.ac.uk/people/tom-crawford For more maths content check out Tom's website https://tomrocksmaths.com/ You can also follow Tom on Facebook, Twitter and Instagram @tomrocksmaths. https://www.facebook.com/tomrocksmaths/ https://twitter.com/tomrocksmaths https://www.instagram.com/tomrocksmaths/ Get your Tom Rocks Maths merchandise here: https://beautifulequations.net/collections/tom-rocks-maths