MATH 294 SPRING 1982 PRELIM 1 # 1
1.6.1 a) Write the system of equations
in the form
b) Find the det A for A in part (a) above.
c) Does A-1 exist?
d) Solve the above system of equations for
A B (i.e. calculate the product AB).
MATH 294 SPRING 1983 FINAL # 10
1.6.2 a) Find a basis for the vector space of all 2x2 matrices.
is the matrix given below,
is an eigenvector of
Find any eigen value of
v = [an eigenvector of
c) Find one solution to each system of equations below, if possible. If not
explain why not.
d) Read carefully. Solve for
in the equation
e) Find the inverse of the matrix
MATH 293 SPRING 1990 PRELIM 1 # 5
1.6.3 a) Find the 2 x 2 matrix A such that if
If B is a matrix and
is a vector, compute
MATH 293 FALL 1991 PRELIM 3 # 3
1.6.4 Write the matrix
as a product of elementary matrices
MATH 293 FALL 1991 PRELIM 3 # 6
1.6.5 Given a matrix A is mxn with m ≠ n and AX = I
and Y A = I (where I is the
corresponding identity matrix), it follow that X = Y . True or false
MATH 293 SPRING 1992 PRELIM 2 # 1
MATH 293 SUMMER 1992 PRELIM 2 # 1
1.6.7 Given three matrices A;B and C,
Find the following products whenever possible
MATH 293 SPRING 1993 PRELIM 2 # 2
1.6.8 Given the matrices
a) Compute BC.
b) Compute CB.
c) Compute (BC - CB)^2.
MATH 293 SPRING 1993 PRELIM 2 # 4
1.6.9 Factor the given matrix, A, a the product LU, where L is a lower
and U is an upper triangular matrix. Label clearly which is L and which is U:
MATH 293 SPRING 1994 PRELIM 2 # 2
1.6.10 (True/false) The following properties hold for the matrix
a) If AM = AN then M = N, where M and N are 3 x 2 matrices.
b) A has an inverse.
c) A is in reduced row echelon form.
d) A is equal to the matrix
e) A and B are row equivalent .
f ) A and B have the same row reduced form.
g) (AT)T = A.
h) BTTA = BAT
MATH 293 FALL 1994 FINAL # 9
1.6.11 Let A and B be n by n matrices. Then
MATH 293 SPRING 1996 PRELIM 2 # 4
We know that Ax = 0: True or false:
1. x is a trivial solution to Ax = 0:
MATH 293 SPRING 1996 PRELIM 2 # 8
1.6.13 Let M be the matrix
Find the entry in the 4th row, 4th column of the matrix M3. Hint: this matrix
the adjacency matrix of a certain graph. Draw the graph and use this to
the entries of M3.
MATH 293 SPRING 1996 FINAL # 34
1.6.14 If A and B are matrices, then (A + B)(A + B) = A2 + 2AB + B2:
True or false.
MATH 293 SPRING 1996 FINAL # 35
1.6.15 If A and B are nxn matrices, then
True or false.
MATH 294 FALL 1997 PRELIM 1 # 1
1.6.16 a) Consider the problem
De termine the general solution to this problem, in vector form.
b) Find a 2 by 2 matrix B, which is not the zero matrix , with B2 = 0.
MATH 294 FALL 1997 FINAL # 4
a) A is the adjacency matrix of a graph G . Find this graph.
b) Using many pencils, we computed the following powers of A :
How many paths of length 10 (i.e. 10 edges) from vertex 2 to vertex 3 are
the graph G?
MATH 294 SPRING 1998 PRELIM 2 # 3
1.6.18 a) All matrices A; B; C; X; Y; Z; and I are nxn and I is the identity
b) Complete the L; U factorization of
MATH 293 SPRING xx xx # 4
1.6.19 a) Find a basis for the row space of the matrix
b) If A is an mxn matrix show that B = AtA and C = AAt are both square. What
are their sizes? Show that B = Bt; C = Ct.
MATH 293 SPRING xx xx # 5
1.6.20 a) If U is an r x m matrix and D and F are r x 1 column vectors,
ex press the
in matrix form.
b) If A is an n x n matrix show that det(cA) = cn det A.
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