In this lecture we use elementary row operations to reduce a matrix to upper triangular form keeping track of how the determinant of the original matrix changes by applying the row operations. We then use the fact that the determinant of an upper triangular matrix is the product of its diagonal entries to compute the determinant of the original matrix. We use the fact that the determinant of a square matrix A is the volume of the box with side vectors the rows of A (up to sign) in order to explain how the determinant changes with elementary row operations. Finally we write down the inverse of any invertible matrix.