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[SOLVED] if 2 columns are identical is there infinite solutions

karush

Well-known member
Jan 31, 2012
3,068
$\tiny{45.4.T40}$
Suppose that the coefficient matrix of a consistent system of linear equations has two columns that are identical. Prove that the system has infinitely many solutions. Refer to the DTSLS Diagram
\item using augmented matrix A for example with c1 and c3 identical
$\left[
\begin{array}{rrr|r}
1&4&1&12\\
2&3&2&14\\
3&2&3&16
\end{array}\right]$
eMH returned the following RREF which show c1 and c3 as pivot columns
this is a violation of RREF
$\text{REFF}(A)=\left[ \begin{array}{rrr|r}
1 & 0 & 1 & 4 \\
0 & 1 & 0 & 2 \\
0 & 0 & 0 & 0
\end{array} \right]$
the matrix was dirived from the the possible set of $x_1=1\ x_2=2\ x_3=3$
a little perplexed as to whar we need to do when one row is all zero's after RREF
also as a result of RREF can this be just an 2x4 augmented matrix, r<n
this is supposed to be a proof which I am not good at
here is the DTSLS Diagram we are supposed to us

03.png
 
Last edited:

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
821
$\left[
\begin{array}{rrr|r}
1&4&1&12\\
2&3&2&14\\
3&2&3&16
\end{array}\right]$
Subtract twice the first row from the second row and three times the first row from the third row:
$\left[\begin{array}{rrr|r} 1&4 & 1 & 12 \\ 0 & -5 & 0 & -10 \\ 0 & -10 & 0 & -20\end{array}\right]$

Divide the second row by -5, subtract 4 times the second row from the first row, and add 10 times the first row to the third row:
$\left[\begin{array}{rrr|r} 1 & 0 & 1 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$
Yes that is what "eMH" (whatever that is) gave you. Now, what does it mean?

Rather than referring to a diagram, write it as "x, y, z" equations.

The top row, "1 0 1 | 4" gives the equation x+ z= 4.
The second row, "0 1 0 | 2" gives the equation y= 2.
The third row, "0 0 0 | 0" gives 0= 0 so no equation at all.

y must equal 2 but x and z can be any pair of numbers that add to 4:
x= 0, y= 2, z= 4
x= 1, y= 2, z= 3
x= 2, y= 2, z= 2
x= 3, y= 2, z= 1
x= 4, y= 2, z= 0
x= 5, y= 2, z= -1
etc.

Yes, there are an infinite number of solutions.
 

karush

Well-known member
Jan 31, 2012
3,068
so is every time we havre r<n by observation we have infinite answers? or does that depend on pivot columns


 
Last edited:

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
821
I have no idea what "r" and "n" mean and they do not appear in your original post.
 

karush

Well-known member
Jan 31, 2012
3,068
r = rows
n = columns

it is usually a very common notation for linear algebra