The augmented matrix of a system of linear equations is made up of the matrix of the coefficients of the unknowns augmented by the matrix of the constants .
Let a
system of linear equations, with the unknowns
, be given by
Write the above system of linear equations using matrices.
The augmented matrix of the above system is given by
Note the vertical separators between the matrix of the coefficient and the matrix of the constants.
is the matrix of the coefficients of the variables and
is the matrix of the constants
Example 1
Write the augmented matrix for the given systems.
Solution to Example 1
The augmented matrices of the given systems of equations are:
Example 2
\( \)\( \)\( \)
Write the systems of equations whose augmented matrices are given below.
a)
\( \begin{bmatrix}
0 & 1 &|& 4\\
-2 & 0 &|& 0
\end{bmatrix}
\)
b)
\( \begin{bmatrix}
-9 & 0 & 0 &|& 4\\
0 & 3 & 0 &|& -1 \\
1 & -3 & 9 &|& -7
\end{bmatrix}
\)
c)
\( \begin{bmatrix}
-4 & 0 & -1 & 0 &|& -4\\
7 & 0 & -2 & 7 &|& -2\\
4 & 0 & 0 & -8 &|& 0 \\
-1 & -5 & 7 & 0 &|& -1
\end{bmatrix}
\)
Solution to Example 2
a)
\( \left\{
\begin{array}{lcl}
y & = & 4 \\
-2x & = & 0
\end{array}
\right.
\)
b)
\( \left\{
\begin{array}{lcl}
-9x & = & 4 \\
3y & = & -1 \\
x -3y + 9z & = & -7
\end{array}
\right.
\)
c)
\( \left\{
\begin{array}{lcl}
-4x - z & = & -4 \\
7x - 2 z + 7 w & = & -2 \\
4x - 8w & = & 0 \\
-x -5 y + 7z & = & -1
\end{array}
\right.
\)
We now show that the three elementary operations on the equations of a system have three equivalent elementary operations on the rows of the augmented matrix of the system.
Operations on Equations | Operations on Rows of Augmented Matrix |
---|---|
|
|
An example that shows that the two types of operations are equivalent is presented below.
Example 3
Use operations on equations and the same operations on rows of the augmented matrix of the system of linear equations given below and show that the systems obtained correspond to the augmented matrices obtained.
\( \left\{
\begin{array}{lcl}
x + y - z & = & -3 \\
-x + 3y + 2z & = & 5 \\
x + 5z & = & 9
\end{array}
\right.
\)
Solution to Example 3
In what follows, equations in the system and rows in the augmented matrix are ranked as follows: equation (1) is the top equation, equation (2) is the second equation from the top and so on and similarly for the rows; row (1) is the top row, row (2) is the second row from the top and so on.
System of Equations | Corresponding Augmented Matrix |
Given: \( \left\{ \begin{array}{lcl} x + y - z & = & -3 \\ -x + 3y + 2z & = & 5 \\ x + 5z & = & 9 \end{array} \right. \) | \(\begin{bmatrix} 1 & 1 & - 1 &|& -3\\ -1 & 3 & 2 &|& 5 \\ 1 & 0 & 5 &|& 9 \end{bmatrix} \) |
add equation (1) to equation (2) | Add row (1) to row (2) |
\( \left\{ \begin{array}{lcl} x + y - z & = & -3 \\ 0x + 4y + z & = & 2 \\ x + 5z & = & 9 \end{array} \right. \) | \( \color{red}{\begin{matrix} \\ R_2 + R_1\\ \\ \end{matrix}} \) \(\begin{bmatrix} 1 & 1 & - 1 &|& -3\\ 0 & 4 & 1 &|& 2 \\ 1 & 0 & 5 &|& 9 \end{bmatrix} \) |
add \( -1 \) times equation (1) to equation (3) | Add \( -1 \) times row (1) to row (3) |
\( \left\{ \begin{array}{lcl} x + y - z & = & -3 \\ 0x + 4y + z & = & 2 \\ 0x - y + 6 z & = & 12 \end{array} \right. \) | \( \color{red}{\begin{matrix} \\ \\ R_3 - R_1\\ \end{matrix}} \) \(\begin{bmatrix} 1 & 1 & - 1 &|& -3\\ 0 & 4 & 1 &|& 2 \\ 0 & -1 & 6 &|& 12 \end{bmatrix} \) |
Interchange equations (2) and (3) | Interchange rows (2) and (3) |
\( \left\{ \begin{array}{lcl} x + y - z & = & -3 \\ 0x - y + 6 z & = & 12\\ 0x + 4y + z & = & 2 \end{array} \right. \) | \( \color{red}{\begin{matrix} \\ R_3\\ R_2\\ \end{matrix}} \) \(\begin{bmatrix} 1 & 1 & - 1 &|& -3\\ 0 & -1 & 6 &|& 12\\ 0 & 4 & 1 &|& 2 \end{bmatrix} \) |
Add 4 times equation (2) to equation (3) | Add 4 times row (2) to row (3) |
\( \left\{ \begin{array}{lcl} x + y - z & = & -3 \\ 0x - y + 6 z & = & 12\\ 0x + 0y + 25z & = & 50 \end{array} \right. \) | \( \color{red}{\begin{matrix} \\ \\ R_3 + 4R_2\\ \end{matrix}} \) \(\begin{bmatrix} 1 & 1 & - 1 &|& -3\\ 0 & -1 & 6 &|& 12\\ 0 & 0 & 25 &|& 50 \end{bmatrix} \) |
Multiply all terms in equation (2) by \( -1 \) | Multiply all terms in row (2) by \( -1 \) |
\( \left\{ \begin{array}{lcl} x + y - z & = & -3 \\ 0x + y - 6 z & = & - 12\\ 0x + 0y + 25z & = & 50 \end{array} \right. \) | \( \color{red}{\begin{matrix} \\ -R_2\\ \\ \end{matrix}} \) \(\begin{bmatrix} 1 & 1 & - 1 &|& -3\\ 0 & 1 & - 6 &|& - 12\\ 0 & 0 & 25 &|& 50 \end{bmatrix} \) |
Multiply all terms in equation (3) by \( \dfrac{1}{25} \) | Multiply all terms in row (3) by \( \dfrac{1}{25} \) |
\( \left\{ \begin{array}{lcl} x + y - z & = & -3 \\ 0x + y - 6 z & = & - 12\\ 0x + 0y + z & = & 2 \end{array} \right. \) | \( \color{red}{\begin{matrix} \\ \\ (1/25)R_3\\ \end{matrix}} \) \(\begin{bmatrix} 1 & 1 & - 1 &|& -3\\ 0 & 1 & - 6 &|& - 12\\ 0 & 0 & 1 &|& 2 \end{bmatrix} \) |
Solutions to the Above Questions