Matrix-Vector Form Write an Augmented Matrix

In summary: So just to clarify, the augmented matrix for this system would be:\begin{array}{ccc}1 & 2 & 1 & 1\\1 & -3 & 3 & 1 \\0 & 4 & -5 & 3 \end{array}In summary, the augmented matrix for this system in vector-matrix form is a 3x4 matrix formed by appending the entries of the column vector b (right hand side of the equation) to those of the coefficient matrix A, with a dotted line separating the fourth column from the rest of the matrix.
  • #1
cosmos42
21
1

Homework Statement


Write in Vector-Matrix form then write the augmented matrix of the system.
r + 2s + t = 1
r - 3s +3t = 1
4s - 5t = 3

Homework Equations


The matrix to which the operations will be applied is called the augmented matrix of the system Ax = b, It is formed by appending the entries of the column vector b (right hand side of the equation) to those of the coefficient matrix A, creating a matrix that is now of order m x (n + 1).

The Attempt at a Solution


I know to use the coefficients to build the rows and columns of a 3 x 3 (?) matrix but I don't understand the augmentation part.
 
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  • #2
cosmos42 said:

Homework Statement


Write in Vector-Matrix form then write the augmented matrix of the system.
r + 2s + t = 1
r - 3s +3t = 1
4s - 5t = 3

Homework Equations


The matrix to which the operations will be applied is called the augmented matrix of the system Ax = b, It is formed by appending the entries of the column vector b (right hand side of the equation) to those of the coefficient matrix A, creating a matrix that is now of order m x (n + 1)^4.
The last bit makes no sense. If matrix A is m x n (m rows by n columns), the augmented matrix will be m x (n + 1), NOT m x (n + 1)^4.
cosmos42 said:

The Attempt at a Solution


I know to use the coefficients to build the rows and columns of a 3 x 3 (?) matrix but I don't understand the augmentation part.
The constants on the right sides of the three equations will be the 4th column of the augmented 3 x 4 matrix.
 
  • #3
Okay, so you have these linear equations:
\begin{array}{lcl}
r + 2s + t & = & 1 \\
r - 3s + 3t & = & 1 \\
4s - 5t & = & 3 \end{array}​
Now, you said you know how to make them into a matrix. The augmentation part is actually really easy. All you have to do is add the answers to the last column.
\begin{array}{ccc}
1 & 2 & 1 & 1\\
1 & -3 & 3 & 1 \\
0 & 4 & -5 & 3\end{array}​
It is common to see a dotted line separating the fourth column from the 3x3 matrix.
 
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  • #4
Mark44 said:
The last bit makes no sense. If matrix A is m x n (m rows by n columns), the augmented matrix will be m x (n + 1), NOT m x (n + 1)^4.

The constants on the right sides of the three equations will be the 4th column of the augmented 3 x 4 matrix.

1.) THE 4 WAS A MISTAKE IT WAS AN INDEX FOR THE FOOTNOTE: "The optional vertical line between the entries of A and those of b emphasizes the way the matrix is constructed"
 
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  • #5
GadgetStrutter said:
Okay, so you have these linear equations:
\begin{array}{lcl}
r + 2s + t & = & 1 \\
r - 3s + 3t & = & 1 \\
4s - 5t & = & 3 \end{array}​
Now, you said you know how to make them into a matrix. The augmentation part is actually really easy. All you have to do is add the answers to the last column.
\begin{array}{ccc}
1 & 2 & 1 & 1\\
1 & -3 & 3 & 1 \\
0 & 4 & -5 & 3\end{array}​
It is common to see a dotted line separating the fourth column from the 3x3 matrix.
Cool thanks!
 
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Related to Matrix-Vector Form Write an Augmented Matrix

1. What is a matrix-vector form?

A matrix-vector form is a way of representing a system of linear equations in terms of matrices and vectors. The matrix contains the coefficients of the variables in the equations, while the vector contains the constants on the right side of the equations.

2. How is a matrix-vector form written?

A matrix-vector form is written as a matrix equation, where the matrix is multiplied by a vector, resulting in another vector. The format is typically [A]x = b, where [A] is the coefficient matrix, x is the vector of variables, and b is the vector of constants.

3. What is an augmented matrix?

An augmented matrix is a representation of a system of linear equations in which the coefficients and constants are combined into a single matrix. The augmented matrix is created by adding a vertical bar between the coefficient matrix and the vector of constants.

4. How is an augmented matrix written?

An augmented matrix is written as [A|b], where [A] is the coefficient matrix and b is the vector of constants. The vertical bar serves as a divider between the two parts of the augmented matrix.

5. What is the purpose of using an augmented matrix?

An augmented matrix is used to simplify and organize a system of linear equations, making it easier to perform operations such as row reduction and finding solutions. It also allows for the use of matrix operations to solve the system, rather than solving each equation individually.

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