Linear Algebra 
Test #2

May 5, 2003

Name____________________ 
R. Hammack

Score ______

(1) This problem concerns the matrix .
(a) Is
in the null space of A?
(b) Is [2 9 7 2] in the row
space of A?
(2)
(3) Find the inverse of this matrix without doing any row reductions.
(Hint: is it orthogonal?)
(4) Suppose . Find
.
(5) Consider the matrix
(a) Find the eigenvalues of A.
(b) Find the eigenspaces of A.
(c) Diagonalize the matrix A, that is find an invertible matrix
P and a diagonal matrix D with .
(6) Suppose is
an orthogonal basis for having
the property that ,
,
and .
Suppose also that satisfies
, ,
and .
Find .
(7) Suppose is
a basis for , and
is a linear transformation for which , ,
and .
Moreover, suppose v is a vector in for
which . Find
T(v).
(8) Suppose A is a matrix which has 5 rows, and the rows are linearly
independent. Suppose also that Null(A) is twodimensional. How many columns
does A have?
(9) Suppose A is a 5×5 matrix with a onedimensional null
space. Find det(A).