WORKSHEET 13 INVERSES OF SQUARE MATRICES In the last ...

WORKSHEET 13
INVERSES OF SQUARE MATRICES
DAVID MEREDITH
In the last worksheet, you showed that if A is a square matrix, then A has a
left inverse if and only if it has a right inverse if and only if the number of rows or
columns is the rank of A. In fact any one-sided inverse of A is the unique two-sided
inverse of A, denoted A −1 .
(1) Show that a square matrix has an inverse if and only if it row reduces to
the identity.
(2) In Octave construct a 5 × 5 matrix A of rank 5.
(3) Construct B = A −1 with the following commands, and show the results are
all the same:
A^(-1), eye(5)/A, A\eye(5), inverse(A), inv(A)
Check that AB = BA = I.
(4) Construct a column vector b with five entries. Solve the equation Ax = b
by calculating x = A −1 b. Show that there are two ways to calculate the
answer with Octave: x = A^(-1)*b, x = A\b. Check that your solution
is correct.
(5) For 2 × 2 matrices there is a simple formula for the inverse, when it exists.
Put Octave aside for([ a moment. ]) a b
(a) Show that rank
= 2 if and only if ad − bc ≠ 0. Argue that
c d
the matrix has rank 2 iff its rows are linearly independent iff the rows
are not parallel iff ad − bc ≠ 0. Fill in the gaps in the argument. The
quantity ad − bc is the determinant of the matrix. (Extra information:
larger square matrices also have determinants, which are 0 precisely
when the matrix is not invertible. However the formula for the determinant
is complex, so it’s best to let Octave calculate determinants of
larger matrices with the command det(A).)
[ ] −1 [ ]
a b 1 d −b
(b) Show that =
.
c d
[
ad
]
− bc −c a
2 3
(c) Find the inverse of and check your answer.
1 4
(6) Suppose you were on a desert island and needed to invert a matrix. Here’s
a procedure you could use. You could also use this procedure in your own
computer program if you needed to invert a matrix.
(a) Construct a 5 × 5 matrix A of rank 5 in Octave. Check that the
row-reduced form of A is I.
(b) Form the double-wide matrix B = [A| I] with the command B=[A,eye(5)].
(c) Row-reduce B to a matrix C: C=rref(B).
Date: March 18, 2010.
1

2 DAVID MEREDITH
(d) You should get C = [I| D]. The matrix D is A −1 . Recover D with
the command: D = C(:,6:10) and check that AD = DA = I.
Department of Mathematics
San Francisco State University
San Francisco, CA 94132
E-mail address: meredith@sfsu.edu
URL: http://online.sfsu.edu/~meredith