question archive State whether each statement is “True” or “False"
State whether each statement is “True” or “False"
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State whether each statement is “True” or “False". Also, Justify each answer. If true, cite appropriate facts/theorems. If it is false, explain why/give a counterexample that shows why the statement is not necessarily true in all cases.
a. Every matrix is row equivalent to a unique matrix in echelon form.
b. Any system of n linear equations in n variables has at most n solutions.
c. If a system of linear equations has two different solu- tions, it must have infinitely many solutions.
d. If a system of linear equations has no free variables, then it has a unique solution.
e. If an augmented matrix A b ] is transformed into [C d ] by elementary row operations, then the equa- tions Ax = b and Cx = d have exactly the same solu- tion sets.
f. If a system Ax = b has more than one solution, then so does the system Ax = 0.
g. If A is an mxn matrix and the equation Ax = b is consistent for some b, then the columns of A span R".
h. If an augmented matrix [A b] can be transformed by elementary row operations into reduced echelon form, then the equation Ax=b is consistent.
i. If matrices A and B are row equivalent, they have the same reduced echelon form.
j. The equation Ax = 0 has the trivial solution if and only if there are no free variables. Home page
k. If A is an mxn matrix and the equation Ax = b is con- sistent for every b in R", then A has m pivot columns.
l. If an mxn matrix A has a pivot position in every row, then the equation Ax=b has a unique solution for each b in R".
m. If an n xn matrix A has n pivot positions, then the reduced echelon form of A is the n x n identity matrix.
n. If 3 x 3 matrices A and B each have three pivot posi- tions, then A can be transformed into B by elementary row operations.
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Answer:
a. False
Every matrix is row equivalent to a unique matrix in row reduced echelon form
b. False
If A has number of equations is less than the number of variables in the echelon form, the system may have infinite number of solutions
if the final equations like : 2x+3y+z=4
3x+4y+2z =5
To solve this we have to give arbitrary value to any one variable. Then we get infinite number of solutions based on the different arbitrary value.
c. True
For a consistent system of equations there are two possibilities, either unique solution or infinite number of solutions
d. False
systems having no free variables may be inconsistent
e. True
Equivalent systems have the same solution
f. True
If AX=b have infinite number of solutions, then AX=0 also have infinite number of solutions
Consistency of a system depends on the rank of coefficient matrix A and augmented matrix Ab and number of varibles
For the system Ax=b to have nfinite number of solution rank of A < no of varibles
This case is satified for AX=0 also
g. False
For the columns of A to span Rm , Ax=b should be consistentfor all b in Rm, not just for one b
h. False
Every system can be transformed to row reduced echelon form , but all systems are not consistent
i. True
Two equivalent matrices have the same row reduced echelon form
j. False
Every homogeneous equation AX= 0 has trivial solution
k. True
A is an m
n matrix and if AX=b is consisten tfor every b, then reducing to echelon form we can see that A has m pivot columns
l.False
The equation has a solution, but it may not be unique
m. True
If A is nn matrix with n pivot positions, then its row reduced echelon matrix will be an identity matrix of order n
n. True.
If A is nn matrix with n pivot positions, then A can reduced to B by elementary row transformations
