If 2 columns are identical is there infinite solutions

In summary, we discussed a system of linear equations with a consistent coefficient matrix and two identical columns. This results in a violation of Reduced Row Echelon Form (RREF) and an infinite number of solutions for the system. We then discussed how this can be represented in the DTSLS Diagram and showed that the system has an infinite number of solutions by converting the augmented matrix to RREF form. We also observed that this is a common occurrence when the number of rows is less than the number of columns.
  • #1
karush
Gold Member
MHB
3,269
5
$\tiny{45.4.T40}$
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:
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  • #2
$\left[
\begin{array}{rrr|r}
1&4&1&12\\
2&3&2&14\\
3&2&3&16
\end{array}\right]$
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.
 
  • #4
I have no idea what "r" and "n" mean and they do not appear in your original post.
 
  • #5
r = rows
n = columns

it is usually a very common notation for linear algebra
 

FAQ: If 2 columns are identical is there infinite solutions

1. What does it mean for 2 columns to be identical?

When 2 columns are identical, it means that every element in one column is exactly the same as the corresponding element in the other column. This means that the columns have the same number of elements and each element has the same value.

2. Can 2 identical columns have infinite solutions?

Yes, 2 identical columns can have infinite solutions. This is because when the columns are identical, any value can be substituted for the variables in the equations and still result in the same solution. Therefore, there are infinite combinations of values that can satisfy the equations.

3. How do you know if 2 columns are identical?

To determine if 2 columns are identical, you can compare each element in one column to the corresponding element in the other column. If all elements are the same, then the columns are identical. Another way to check is by solving the equations represented by the columns and seeing if they result in the same solution.

4. Can 2 columns be identical but have a different number of rows?

No, for 2 columns to be identical, they must have the same number of rows. This is because each element in one column must have a corresponding element in the same position in the other column. If the columns have a different number of rows, then there will be elements without a corresponding element, making the columns not identical.

5. What is the significance of 2 columns being identical in mathematics?

When 2 columns are identical, it means that the equations represented by the columns are equivalent. This can be useful in solving systems of equations, as it means that the equations have the same solution. It also allows for easier manipulation and simplification of equations.

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