id: 05838514
dt: j
an: 2011a.00749
au: Stuart, Jeffrey
ti: Building generalized inverses of matrices using only row and column
operations.
so: Int. J. Math. Educ. Sci. Technol. 41, No. 8, 1102-1113 (2010).
py: 2010
pu: Taylor \& Francis, Abingdon, Oxfordshire
la: EN
cc: H65 N35
ut: matrix inversion; elementary row operation; Gauss-Jordan algorithm;
generalized inverse; Moore-Penrose inverse
ci:
li: doi:10.1080/0020739X.2010.500696
ab: Summary: Most students complete their first and only course in linear
algebra with the understanding that a real, square matrix $A$ has an
inverse if and only if $\mathrm{rref}(A)$, the reduced row echelon form
of $A$, is the identity matrix $I_n$. That is, if they apply elementary
row operations via the Gauss-Jordan algorithm to the partitioned matrix
$[A | I_n]$ to obtain $[\mathrm{rref}(A) | P]$, then the matrix $A$ is
invertible exactly when $\mathrm{rref}(A) = I_n$, in which case, $P =
A^{-1}$. Many students must wonder what happens when $A$ is not
invertible, and what information $P$ conveys in that case. That
question is, however, seldom answered in a first course. We show that
investigating that question emphasizes the close relationships between
matrix multiplication, elementary row operations, linear systems, and
the four fundamental spaces associated with a matrix. More important,
answering that question provides an opportunity to show students how
mathematicians extend results by relaxing hypotheses and then exploring
the strengths and limitations of the resulting generalization, and how
the first relaxation found is often not the best relaxation to be
found. Along the way, we introduce students to the basic properties of
generalized inverses. Finally, our approach should fit within the time
and topic constraints of a first course in linear algebra.
rv: