Matrix Operations

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

e) Let Find 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.
b) is the matrix given below, is an eigenvector of . Find any eigen value of .

with v = [an eigenvector of

c) Find one solution to each system of equations below, if possible. If not possible,
explain why not.

d) Read carefully. Solve for in the equation with:

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 whenever possible.

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

1.6.6

MATH 293 SUMMER 1992 PRELIM 2 # 1

1.6.7 Given three matrices A;B and C,

Find the following products whenever possible
a) AB
b) BA
c) A(BC)
d) CB

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 triangular matrix,
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) (A^T)^T = A.
h) BT^TA = BA^T

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

1.6.12 Let

Let   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 is
the adjacency matrix of a certain graph. Draw the graph and use this to interpret
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 where

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

1.6.17 Let

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 there in
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 matrix. The
inverse of

Find X.
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 = A^tA and C = AA^t are both square. What
are their sizes? Show that B = B^t; C = C^t.

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
relation

in matrix form.

b) If A is an n x n matrix show that det(cA) = cⁿ det A.