# La1.1.22 3 aug matrices for linear sys solution set is x_1=3,x_2=-2,x_3=-1

• MHB
• karush
In summary: It should be:$$A_3=\left[\begin{array}{rrrrr} 0& 0& 2& -2\\ 0& 1& 2& -4\\ 1& 0& 0& 3\\ \end{array}\right]$$In summary, we discussed constructing 3 different augmented matrices for linear systems whose solution set is $x_1=3, x_2=-2, x_3=-1$. The simplest augmented matrix would follow from the equations $x_1=3, x_2=-2, x_3=-1$ and would be a $3\times4$ matrix. We also explored the concept of augmented matrices
karush
Gold Member
MHB
$$\tiny{la1.1.22}$$

$\textsf{Construct 3 different augmented matrices for linear systems whose solution set is}$
$$x_1=3,\quad x_2=-2,\quad x_3=-1$$Well we could start just by$$3x-2y+y=-1$$but then we need a $3\times4$ matrix

karush said:
$$\tiny{la1.1.22}$$

$\textsf{Construct 3 different augmented matrices for linear systems whose solution set is}$
$$x_1=3,\quad x_2=-2,\quad x_3=-1$$

Well we could start just by
$$3x-2y+y=-1$$

When we substitute $x=3$ and $y=-2$, we get $3\cdot 3 - 2\cdot (-2) + (-2)=11 \ne -1$.
That's not going to work is it?

karush said:
but then we need a $3\times4$ matrix

We need indeed a 3x4 matrix since we have 3 unknowns, meaning we should have 3 equations for them.
The simplest augmented matrix would follow from:
$$\begin{cases}x_1=3 \\ x_2=-2 \\ x_3=-1\end{cases}$$
What is the corresponding augmented matrix?

I like Serena said:
When we substitute $x=3$ and $y=-2$, we get $3\cdot 3 - 2\cdot (-2) + (-2)=11 \ne -1$.
That's not going to work is it?

well actually its $x+y+z=3-2-1=0$
I like Serena said:
We need indeed a 3x4 matrix since we have 3 unknowns, meaning we should have 3 equations for them.
The simplest augmented matrix would follow from:
$$\begin{cases}x_1=3 \\ x_2=-2 \\ x_3=-1\end{cases}$$
What is the corresponding augmented matrix?

ok presumably this would be one case

$$\left[\begin{array}{rrrrr} %x_1& x_2& x_3& 0\\ x_1& 0& 0& 3\\ 0& x_2& 0& -2\\ 0& 0& x_3& -1\\ \end{array}\right]$$

Last edited:
karush said:
ok presumably this would be one case

$$\left[\begin{array}{rrrrr} %x_1& x_2& x_3& 0\\ x_1& 0& 0& 3\\ 0& x_2& 0& -2\\ 0& 0& x_3& -1\\ \end{array}\right]$$

... except that an augmented matrix is without the variables.
So it should be:
$$\left[\begin{array}{rrrrr} %x_1& x_2& x_3& 0\\ 1& 0& 0& 3\\ 0& 1& 0& -2\\ 0& 0& 1& -1\\ \end{array}\right]$$

And if we add both the second row and the third row to the first row, the first row becomes the equation that you've suggested... and we have a new augmented matrix.

I like Serena said:
... except that an augmented matrix is without the variables.
So it should be:
$$\left[\begin{array}{rrrrr} %x_1& x_2& x_3& 0\\ 1& 0& 0& 3\\ 0& 1& 0& -2\\ 0& 0& 1& -1\\ \end{array}\right]$$

And if we add both the second row and the third row to the first row, the first row becomes the equation that you've suggested... and we have a new augmented matrix.

the only other augmented matrix I see is just flipping this one to$$\left[\begin{array}{rrrrr} 0& 0& 1& -1\\ 0& 1& 0& -2\\ 1& 0& 0& 3\\ \end{array}\right]$$so the the third ?

karush said:
the only other augmented matrix I see is just flipping this one to$$\left[\begin{array}{rrrrr} 0& 0& 1& -1\\ 0& 1& 0& -2\\ 1& 0& 0& 3\\ \end{array}\right]$$so the the third ?

Do the reverse of a row reduction?
That is, add/substract rows such that instead of elements becoming zero, they become non-zero?

I like Serena said:
Do the reverse of a row reduction?
That is, add/substract rows such that instead of elements becoming zero, they become non-zero?
i thot augmented meant you had to have a triangle of zeros

karush said:
i thot augmented meant you had to have a triangle of zeros

Neh, an augmented matrix is just an encoding of a set of equations into a single matrix.
When we apply row reduction to an augmented matrix, we get a new augmented matrix in row reduced echelon form, which has indeed a triangle of zeroes.

I like Serena said:
Neh, an augmented matrix is just an encoding of a set of equations into a single matrix.
When we apply row reduction to an augmented matrix, we get a new augmented matrix in row reduced echelon form, which has indeed a triangle of zeroes.

ok here goes...

$$A=\left[\begin{array}{rrrrr} 0& 0& 1& -1\\ 0& 1& 0& -2\\ 1& 0& 0& 3\\ \end{array}\right]$$
$\textsf{so then if from$A$we multiply$R_1$by$2$and add to$R_3$to become$A_2$}$
$$A_2=\left[\begin{array}{rrrrr} 0& 0& 2& -2\\ 0& 1& 0& -2\\ 1& 0& 2& 1\\ \end{array}\right]$$
$\textsf{so then if from$A$we multiply$R_1$by$2$and add to$R_2$to become$A_3$}$
$$A_3=\left[\begin{array}{rrrrr} 0& 0& 2& -2\\ 0& 1& 2& 0\\ 1& 0& 0& 3\\ \end{array}\right]$$so then thusly
kinda maybe hopefully?

karush said:
ok here goes...

$$A=\left[\begin{array}{rrrrr} 0& 0& 1& -1\\ 0& 1& 0& -2\\ 1& 0& 0& 3\\ \end{array}\right]$$
$\textsf{so then if from$A$we multiply$R_1$by$2$and add to$R_3$to become$A_2$}$
$$A_2=\left[\begin{array}{rrrrr} 0& 0& 2& -2\\ 0& 1& 0& -2\\ 1& 0& 2& 1\\ \end{array}\right]$$
$\textsf{so then if from$A$we multiply$R_1$by$2$and add to$R_2$to become$A_3$}$
$$A_3=\left[\begin{array}{rrrrr} 0& 0& 2& -2\\ 0& 1& 2& 0\\ 1& 0& 0& 3\\ \end{array}\right]$$so then thusly
kinda maybe hopefully?

All good except for $A_3$ where there is a mistake with a sign.

## What is the meaning of "La1.1.22 3 aug matrices for linear sys solution set is x_1=3,x_2=-2,x_3=-1"?

The notation "La1.1.22 3 aug matrices for linear sys solution set is x_1=3,x_2=-2,x_3=-1" refers to a linear system of equations that can be represented using augmented matrices. The solution set for this system is given by x_1=3, x_2=-2, and x_3=-1.

## What is an augmented matrix?

An augmented matrix is a representation of a linear system of equations in matrix form, where the coefficients of the variables are placed in a rectangular matrix and the constants are placed in a column vector on the right side of the matrix.

## How is the solution set for a linear system of equations found using augmented matrices?

The solution set for a linear system of equations can be found by performing elementary row operations on the augmented matrix until it is in reduced row echelon form. The resulting matrix will have the solutions to the system of equations on the right side.

## What does it mean for a matrix to be in reduced row echelon form?

A matrix is in reduced row echelon form when it has the following properties: 1) all rows of zeros are at the bottom, 2) the first non-zero entry in each row (called a leading entry) is a 1, 3) each leading entry is the only non-zero entry in its column, and 4) the leading entry of each row is to the right of the leading entry in the row above it.

## How do the solutions in an augmented matrix correspond to the solutions to the linear system of equations?

The solutions in an augmented matrix correspond to the solutions to the linear system of equations by using the values in the rightmost column as the values for the variables in the system. In the example provided, x_1=3, x_2=-2, and x_3=-1 would be the solutions to the system of equations.

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