Determinants By Way Of Gauss-Jordan Reduction

Mathematics – Linear algebra I: matrix algebra Determinants by way of Gauss-Jordan reduction Given a square matrix A, we can compute a number called the determinant of A, usually denoted by |A| or det(A), that gives a lot of information about A. For example, |A| 6= 0 exactly when A1 exists. One problem with the usual definition of determinants [see §4.2 in the text], which works by reducing the determinant of an n ⇥ n matrix to an alternating sum of determinants of n di↵erent (n 1) ⇥ (n 1) sub-matrices, is that computing them this way is a lot of work unless A is a pretty small matrix or has a lot of 0s. (Heck, it’s a pain even for 3 ⇥ 3 matrices with the usual definition, as we saw in computing cross-products of vectors in R3.) In this assignment, we will be looking at a method to compute the determinant of a matrix using the Gauss-Jordan method. The determinant of an n ⇥ n matrix A satisfies the following rules: i. The identity matrix has determinant equal to 1, i.e. |In| = 1. ii. If you exchange the ith and jth row of A to get the matrix B, then |B| = |A|. iii. If you multiply the ith row of A by a constant c to get the matrix C, then |C| = c|A|. iv. If you add a multiple of any row of A to a di↵erent row of A to get the matrix D, then |D| = |A|. v. Taking the transpose of A doesn’t change the determinant. That is, |AT | = |A|. If you really wanted to, by the way, you could actually use this collection of rules as the definition of the determinant of a matrix. It’s pretty cumbersome as a definition, but it does provide a much more ecient way to compute the determinant of even a modestly large matrix. It also makes it easier to see why A is invertible if and only if |A| 6= 0: both are equivalent to the matrix being reducible to In using the Gauss-Jordan method. 1. In both a and b use the Gauss-Jordan method to put the matrix A in reduced rowechelon form, and then apply rules i–v to work out |A|. a. A = 3 2 2 4 [2] b. A = 2 4 103 354 261 3 5 [3] 2. Use rules i–v to determine |A| if: a. A = O = 0 0 0 0 . [1] b. A has a row of zeros. [1] c. A has two equal rows. [1] 3. Rules ii – iv are true for the columns of A as well as the rows. Explain why. [2]