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 two-dimensional. How many columns does A have?

(9) Suppose A is a 5×5 matrix with a one-dimensional null space.   Find det(A).