Question
In: Advanced Math
prove the following statement: If the augmented matrices of two linear systems are row equivalent, then...
prove the following statement: If the augmented matrices of two linear systems are row equivalent, then those systems are equivalent.
(To do this, start with a solution to one of the systems and show that it is still a solution of the other system under each of the three elementary row operations.)
Solutions
Colby Messinger answered 1 year agoRelated Solutions
please give examples of four row equivalent matrices which are all in row echelon forms (not...
please give examples of four row equivalent matrices which are all
in row echelon forms (not necessarly reduced row echelon)
Write the following matrices into row echelon form.
Exercise1. Write the following matrices into row echelon form.
1---- . Is the statement "Elementary row operations on an augmented matrix never change the solution...
1----
. Is the statement "Elementary row operations on an augmented
matrix never change the solution set of the associated linear
system" true or false? Explain.
A.
True, because elementary row operations are always applied to
an augmented matrix after the solution has been found.
B.
False, because the elementary row operations make a system
inconsistent.
C.
True, because the elementary row operations replace a system
with an equivalent system.
D.
False, because the elementary row operations augment the number...
Exercise1. Write the following matrices into row echelon form.
Exercise1. Write the following matrices into row echelon form. (a) A=⎛⎜⎝1132−22100474⎞⎟⎠A=(1132−22100474) (c) C=⎛⎜⎝121457210121331578⎞⎟⎠C=(121457210121331578) (b) B=⎛⎜⎝234304131⎞⎟⎠B=(234304131) (d) D=⎛⎜ ⎜ ⎜⎝11111−11−1−1−111−11λλ⎞⎟ ⎟ ⎟⎠
Suppose R and R0 are 2 ⇥ 3 row-reduced echelon matrices and that the systems RX...
Suppose R and R0 are 2 ⇥ 3 row-reduced echelon matrices and that
the systems RX = 0
and
R’X = 0 have exactly the same solutions. Prove that R = R’
.
The following matrix is the augmented matrix for a system of linear equations. A = 1...
The following matrix is the augmented matrix for a system of
linear equations. A =
1
1
0
1
1
0
0
1
3
3
0
0
0
1
1
2
2
0
5
5
(a) Write down the linear system of equations whose augmented
matrix is A.
(b) Find the reduced echelon form of A.
(c) In the reduced echelon form of A, mark the pivot
positions.
(d) Does the system have no solutions, exactly one solution or
infinitely...
Show that two m×n matrices are equivalent if and only if they have the same invariant...
Show that two m×n matrices are equivalent if and only if they
have the same invariant factors, i.e. (by Problem 4), if and only
if they have the same Smith normal form.
For each of the following matrices, find a minimal spanning set for its Column space, Row...
For each of the following matrices, find a minimal spanning set
for its Column space, Row space,and Nullspace. Use Octave Online to
get matrix A into RREF.
A = [4 6 10 7 2; 11 4 15 6 1; 3 −9 −6 5 10]
Find the reduced row echelon form of the following matrices. Interpret your result by giving the...
Find the reduced row echelon form of the following matrices.
Interpret your result by giving the solutions of the systems whose
augmented matrix is the one given.
[ 0 0 3 -1 5
1 0 0 4 2
4 1 3 0 -8
1 2 7 9 0 ]
Prove that the following two statements are not logically equivalent. In your proof, completely justify your...
Prove that the following two statements are not logically
equivalent. In your proof, completely justify your answer.
(a) A real number is less than 1 only if its reciprocal is
greater than 1.
(b) Having a reciprocal greater than 1 is a sufficient condition
for a real number to be less than 1.
Proof:
#2.
Prove that the following is a valid argument:
All real numbers
have nonnegative squares.
The number i has a negative square.
Therefore, the...
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