Review exercise 1.26. (a) Given a 3 by 3 matrix what vector
would make the product
have 1 times column 1 of A plus 2 times column 3?
(b) Construct a matrix for which the sum of column 1 and 2 times column 3 is zero, and show that
is singular. Why?
Answer: (a) If we choose
then will consist of column 1 of
plus 2 times column 3. In essence if
then
(b) The following matrix has column 1 plus 2 times column 3 equal to 0.
Elimination of this matrix proceeds as follows:
Since there is no pivot in row 3 the matrix is singular. The problem is that since column 3 is a multiple of column 1 (being equal to -2 times column 1) the elimination steps that produce zeros in the and
position will also produce zeros in the
and
positions, so that there is no possible pivot in column 3.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition
and the accompanying free online course, and Dr Strang’s other books
.