Question

For each statement below, either show that the statement is true or give an example showing that it is false. Assume throughout that A and B are square matrices, unless otherwise specified.

(a) If AB = 0 and A ̸= 0, then B = 0.

(b) If x is a vector of unknowns, b is a constant column vector,
and Ax = b has no solution, then Ax = 0 has no solution.

(c) If x is a vector of unknowns and Ax = 0 has only the trivial solution, then, for every column vector b, the system Ax = b has a unique solution.

(d) If a system of simultaneous linear equations has more than one solution, then the reduced row echelon form of the system must have a row of zeroes.

(e) If in a system of simultaneous linear equations, there are more variables than equations, then there are infinitely many solutions.

Answer #1

(a). FALSE. If A =

1 |
0 |

0 |
0 |

and B =

0 |
0 |

1 |
0 |

then AB = 0, A≠0 and B≠0.

(b). FALSE. x = 0 i.e. the trivial solution is always a solution of the homogeneous equation Ax = 0.

( c). TRUE. If x = 0 is the only solution of the equation Ax =
0, then A has full rank so that A is invertible. Then x =
A^{-1} b is the unique solution to Ax = b regardless of the
choice of b.

(d). TRUE. If the RREF of A does not have any zero row, then A
has full rank so that A is invertible. Then x = A^{-1} b is
the unique solution to Ax = b. Therefore, if there is more than one
solution, then the RREF of A must have a zero row.

(e). TRUE. In such a case, there will be some free variables so that there are infinitely many solutions.

Is each statement true or false? If true, explain why; if false,
give a counterexample.
a) A linear system with 5 equations and 4 unknowns is always
inconsistent.
b) If the coefficient matrix of a homogeneous system has a
column of zeroes, then the system has infinitely many solutions.
(Note: a homogeneous system has augmented matrix [A | b] where b =
0.)
c) If the RREF of a homogeneous system has a row of zeroes, then
the system has...

Choose either true or false for each
statement
a. There is a vector [b1 b2] so that the set of solutions to
1
0
1
0
1
0
[ x1, x2 , x3,] =[b1b2] is the z-axis.
b. The homogeneous system Ax=0 has the trivial solution if and
only if the system has at least one free variable.
c. If x is a nontrivial solution of Ax=0, then every entry of x
is nonzero.
d. The equation Ax=b is homogeneous...

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5
0 0]
b) A vector b is a linear combination of the columns of a matrix A
if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u =
+ p + vh
where vh is any solution...

Consider a system of linear equations with augmented matrix A
and coefficient matrix C. In each case
either prove the statement or give an example showing that it is
false.
• If there is more than one solution, A has a row of
zeros.
• If A has a row of zeros, there is more than one solution.
• If there is no solution, the row-echelon form of C has a row of
zeros. • If the row-echelon form of...

Given that A and B are n × n matrices and T is a linear
transformation. Determine which of the following is FALSE.
(a) If AB is not invertible, then either A or B is not
invertible.
(b) If Au = Av and u and v are 2 distinct vectors, then A is not
invertible.
(c) If A or B is not invertible, then AB is not invertible.
(d) If T is invertible and T(u) = T(v), then u =...

In each case below show that the statement is True or give an
example showing that it is False.
(i) If {X, Y } is independent in R n, then {X, Y, X + Y } is
independent.
(ii) If {X, Y, Z} is independent in R n, then {Y, Z} is
independent.
(iii) If {Y, Z} is dependent in R n, then {X, Y, Z} is
dependent.
(iv) If A is a 5 × 8 matrix with rank A...

Find the values of a and b for which the following system of
linear equations is (i) inconsistent; (ii) has a unique solution;
(iii) has infinitely many solutions. For the case where the system
has infinitely many solutions, write the general solution.
x + y + z = a
x + 2y ? z = 0
x + by + 3z = 2

q.1.(a)
The following system of linear equations has an infinite number
of solutions
x+y−25 z=3
x−5 y+165 z=0
4 x−14 y+465 z=3
Solve the system and find the solution in the form
x(vector)=ta(vector)+b(vector)→, where t is a free
parameter.
When you write your solution below, however, only write the part
a(vector=⎡⎣⎢ax ay az⎤⎦⎥ as a unit column vector with the
first coordinate positive. Write the results accurate to
the 3rd decimal place.
ax =
ay =
az =

Use Gauss-Jordan method (augmented matrix method) to
solve the following systems of linear equations.
Indicate whether the system has a unique solution, infinitely many
solutions, or no solution. Clearly write
the row operations you use.
(a)
x − 2y + z = 8
2x − 3y + 2z = 23
− 5y + 5z = 25
(b)
x + y + z = 6
2x − y − z = 3
x + 2y + 2z = 0

7. Answer the following questions true or false and provide an
explanation. • If you think the statement is true, refer to a
definition or theorem. • If false, give a counter-example to show
that the statement is not true for all cases.
(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then
the equation Ax = b has a unique solution for all b in R^3 .
(b) If the augmented...

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