Determinants

In the previous chapters, we pointed out the importance of having a clear criterion by which we can determine whether a matrix is non-singular, and if so, the existence of a method by which we can calculate its inverse. The term determinant plays a key role in this. There are different approaches to the definition of the determinant, but here we show the approach where the determinant is defined by induction. In doing so, we introduce the concepts of minor, cofactor and Laplace expansion. Also, we state the Sarrus rule for calculating determinants of the order 3. After giving its geometrical interpretation, we state the Binet-Cauchy theorem and apply it to derive a criterion on how to use the determinant to check if a matrix is non-singular. We also give the formula for the inverse of a matrix by using the adjoint, which is the transpose of the corresponding matrix of cofactors. Finally, we present Cramer’s rule for solving Cramer’s system, that is a system AX=B in which the number of equations is equal to the number of unknowns and matrix A is non-singular, as well as give two economic applications of Cramer’s rule—determining the equilibrium price in a market with multiple goods, the IS-LM model and the Keynesian model of income determination.