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 =
b. A =
2. Use rules i–v to determine |A| if:
a. A = O =
b. A has a row of zeros. 
c. A has two equal rows. 
3. Rules ii – iv are true for the columns of A as well as the rows. Explain why. 
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