August 7, 2025
Examples: Do your best to describe all of the solutions to each of the following linear systems.
Reminder: There are three possibilities for the solution set of a linear system.
No other alternatives are possible.
Goal for Today: Develop a framework which can be used to investigate systems, arrive at solutions, and to describe those solution spaces.
Operations you’ve made use of, whether you noticed or not…
None of these operations change the solution set to a system.
Working with systems in their raw forms, as we’ve seen them so far, makes it difficult to develop and apply a standard algorithm for solving systems.
\(\bigstar\) Encoding these systems via matrices will afford us this opportunity.
Definition (Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
Definition (Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its coefficient matrix is defined as the \(m\times n\) matrix
Definition (Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its coefficient matrix is defined as the \(m\times n\) matrix
\[\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right]\]
Definition (Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its coefficient matrix is defined as the \(m\times n\) matrix
\[\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right]\]
Example (Coefficient Matrix): For example, the coefficient matrix corresponding to the linear system \(\left\{\begin{array}{rcr} 2x_1 + 5x_2 & = & 13\\ -2x_1 + 3x_2 & = & 11\end{array}\right.\) is…
Definition (Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its coefficient matrix is defined as the \(m\times n\) matrix
\[\left[\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{array}\right]\]
Example (Coefficient Matrix): For example, the coefficient matrix corresponding to the linear system \(\left\{\begin{array}{rcr} 2x_1 + 5x_2 & = & 13\\ -2x_1 + 3x_2 & = & 11\end{array}\right.\) is \(\left[\begin{array}{cc} 2 & 5\\ -2 & 3\end{array}\right]\).
The coefficient matrix on its own is not enough to encode our systems. We’ll almost always be interested in the augmented coefficient matrix instead.
Definition (Augmented Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its augmented coefficient matrix is defined as
\[\left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \end{array}\right]\]
Example (Coefficient Matrix): For example, the augmented coefficient matrix corresponding to the linear system \(\left\{\begin{array}{rcr} 2x_1 + 5x_2 & = & 13\\ -2x_1 + 3x_2 & = & 11\end{array}\right.\) is \(\left[\begin{array}{cc|c} 2 & 5 & 13\\ -2 & 3 & 11\end{array}\right]\).
The coefficient matrix on its own is not enough to encode our systems. We’ll almost always be interested in the augmented coefficient matrix instead.
Definition (Augmented Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
The coefficient matrix on its own is not enough to encode our systems. We’ll almost always be interested in the augmented coefficient matrix instead.
Definition (Augmented Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its augmented coefficient matrix is defined as
The coefficient matrix on its own is not enough to encode our systems. We’ll almost always be interested in the augmented coefficient matrix instead.
Definition (Augmented Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its augmented coefficient matrix is defined as
\[\left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \end{array}\right]\]
The coefficient matrix on its own is not enough to encode our systems. We’ll almost always be interested in the augmented coefficient matrix instead.
Definition (Augmented Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its augmented coefficient matrix is defined as
\[\left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \end{array}\right]\]
Example (Coefficient Matrix): For example, the augmented coefficient matrix corresponding to the linear system \(\left\{\begin{array}{rcr} 2x_1 + 5x_2 & = & 13\\ -2x_1 + 3x_2 & = & 11\end{array}\right.\) is…
The coefficient matrix on its own is not enough to encode our systems. We’ll almost always be interested in the augmented coefficient matrix instead.
Definition (Augmented Coefficient Matrix): Given a system
\[\left\{\begin{array}{lcl} a_{11}x_1 + a_{12}x_2 + \cdots a_{1n}x_n & = & b_1\\ a_{21}x_1 + a_{22}x_2 + \cdots a_{2n}x_n & = & b_1\\ & \vdots & \\ a_{m1}x_1 + a_{m2}x_2 + \cdots a_{mn}x_n & = & b_m \end{array}\right.\]
its augmented coefficient matrix is defined as
\[\left[\begin{array}{cccc|c} a_{11} & a_{12} & \cdots & a_{1n} & b_1\\ a_{21} & a_{22} & \cdots & a_{2n} & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ a_{m1} & a_{m2} & \cdots & a_{mn} & b_m \end{array}\right]\]
Example (Coefficient Matrix): For example, the augmented coefficient matrix corresponding to the linear system \(\left\{\begin{array}{rcr} 2x_1 + 5x_2 & = & 13\\ -2x_1 + 3x_2 & = & 11\end{array}\right.\) is \(\left[\begin{array}{cc|c} 2 & 5 & 13\\ -2 & 3 & 11\end{array}\right]\).
Definition (Elementary Row Operations): Any of the following elementary row operations can be done on an augmented coefficient matrix corresponding to a system of linear equations without changing the solution set.
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
(Hint. Try to obtain an augmented coefficient matrix of the form \(\left[\begin{array}{ccc|c} \blacksquare & * & * & *\\ 0 & \blacksquare & * & *\\ 0 & 0 & \blacksquare & *\end{array}\right]\))
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution. We’ll start with the corresponding augmented coefficient matrix and then row-reduce it.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 6 & 2 & 1 & 10\\ 0 & -4 & 1 & -2\end{array}\right] \end{align*}\]
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution. We’ll start with the corresponding augmented coefficient matrix and then row-reduce it.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 6 & 2 & 1 & 10\\ 0 & -4 & 1 & -2\end{array}\right] &\substack{R_2 \leftarrow R_2 + \left(-3\right)R_1\\ \longrightarrow} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 5 & -8 & -11\\ 0 & -4 & 1 & -2\end{array}\right] \end{align*}\]
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution. We’ll start with the corresponding augmented coefficient matrix and then row-reduce it.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 5 & -8 & -11\\ 0 & -4 & 1 & -2\end{array}\right] \end{align*}\]
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution. We’ll start with the corresponding augmented coefficient matrix and then row-reduce it.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 5 & -8 & -11\\ 0 & -4 & 1 & -2\end{array}\right] &\substack{R_2 \leftarrow 4R_2\\ R_3 \leftarrow 5R_3\\ \longrightarrow} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 20 & -32 & -44\\ 0 & -20 & 5 & -10\end{array}\right] \end{align*}\]
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution. We’ll start with the corresponding augmented coefficient matrix and then row-reduce it.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 20 & -32 & -44\\ 0 & -20 & 5 & -10\end{array}\right] \end{align*}\]
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution. We’ll start with the corresponding augmented coefficient matrix and then row-reduce it.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 20 & -32 & -44\\ 0 & -20 & 5 & -10\end{array}\right] &\substack{R_3 \leftarrow R_3 + R_2\\ \longrightarrow} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 20 & -32 & -44\\ 0 & 0 & -27 & -54\end{array}\right]\\ \end{align*}\]
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 6 & 2 & 1 & 10\\ 0 & -4 & 1 & -2\end{array}\right] &\sim \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 20 & -32 & -44\\ 0 & 0 & -27 & -54\end{array}\right]\\ \end{align*}\]
From here, the bottom row of the augmented coefficient matrix gives \(-27x_3 = -54\), so \(\boxed{~\displaystyle{x_3 = 2}~}\).
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 6 & 2 & 1 & 10\\ 0 & -4 & 1 & -2\end{array}\right] &\sim \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 20 & -32 & -44\\ 0 & 0 & -27 & -54\end{array}\right]\\ \end{align*}\]
From here, the bottom row of the augmented coefficient matrix gives \(-27x_3 = -54\), so \(\boxed{~\displaystyle{x_3 = 2}~}\). The second row now tells us that \(20x_2 - 32\left(2\right) = -44\), so \(\boxed{~\displaystyle{x_2 = 1}~}\).
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 6x_1 + 2x_2 + x_3 & = & 10\\ -4x_2 + x_3 & = & -2\end{array}\right.\) and use elementary row operations to solve the system.
Solution.
\[\begin{align*} \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 6 & 2 & 1 & 10\\ 0 & -4 & 1 & -2\end{array}\right] &\sim \left[\begin{array}{rrr|r} 2 & -1 & 3 & 7\\ 0 & 20 & -32 & -44\\ 0 & 0 & -27 & -54\end{array}\right]\\ \end{align*}\]
From here, the bottom row of the augmented coefficient matrix gives \(-27x_3 = -54\), so \(\boxed{~\displaystyle{x_3 = 2}~}\). The second row now tells us that \(20x_2 - 32\left(2\right) = -44\), so \(\boxed{~\displaystyle{x_2 = 1}~}\). Finally, the first row tells us that \(2x_1 - (1) + 3(2) = 7\), giving that \(\boxed{~\displaystyle{x_1 = 1}~}\). \(_\blacktriangledown\)
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcl} 2x_1 + 4x_2 &= & 10\\ x_1 - 3x_2 &= & 15\end{array}\right.\) and solve the system using elementary row operations.
(Hint. Try to obtain an augmented coefficient matrix of the form \(\left[\begin{array}{cc|c} \blacksquare & * & *\\ 0 & \blacksquare & *\end{array}\right]\))
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ -4x_1 + x_2 + x_3 & = & -1\\ -4x_2 + x_3 & = & 10\end{array}\right.\) and use elementary row operations to solve the system.
(Hint. Try to obtain an augmented coefficient matrix of the form \(\left[\begin{array}{ccc|c} \blacksquare & * & * & *\\ 0 & \blacksquare & * & *\\ 0 & 0 & \blacksquare & *\end{array}\right]\))
Example: Construct the augmented coefficient matrix corresponding to the system \(\left\{\begin{array}{rcr}2x_1 - x_2 + 3x_3 & = & 7\\ 4x_1 + x_2 +x_3 & = & 7\\ 4x_1 - 2x_2 + 6x_3 & = & 14\end{array}\right.\) and use elementary row operations to solve the system.
(Hint. Try to obtain an augmented coefficient matrix of the form \(\left[\begin{array}{ccc|c} \blacksquare & * & * & *\\ 0 & \blacksquare & * & *\\ 0 & 0 & \blacksquare & *\end{array}\right]\))
\[\left[\begin{array}{cccc|c} \blacksquare & * & \cdots & * & b_1\\ 0 & \blacksquare & \cdots & * & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & \blacksquare & b_n\end{array}\right]\]
The form shown above is called row echelon form
From this form, we can use back substitution to solve the corresponding system
\[\left[\begin{array}{cccc|c} \blacksquare & * & \cdots & * & b_1\\ 0 & \blacksquare & \cdots & * & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & \blacksquare & b_n\end{array}\right]\]
In REF, the black squares (\(\blacksquare\)) denote the first non-zero entry in each row
All entries below these pivots are \(0\)’s, but values above pivots are permitted to be anything.
Example: All of the following matrices are in row echelon form. The pivot entries are boxed up for convenience.
\[A = \left[\begin{array}{cc|c} \boxed{~8~} & 2 & -4\\ 0 & \boxed{~3~} & 5\end{array}\right]~~~~~B = \left[\begin{array}{cccc|c} \boxed{~1~} & -4 & 1 & 9 & -2\\ 0 & 0 & \boxed{~3~} & 1 & -1\\ 0 & 0 & 0 & \boxed{~2~} & 4\end{array}\right]\]
\[C = \left[\begin{array}{ccc|c} \boxed{~4~} & -2 & 9 & -1\\ 0 & \boxed{~4~} & 1 & -4\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\end{array}\right]~~~~~D = \left[\begin{array}{ccc|c} \boxed{~7~} & 2 & 1 & 5\\ 0 & 0 & 0 & \boxed{~-3~}\end{array}\right]\]
Note: It is not necessarily the case that every diagonal element contains a pivot.
Example: The matrix \(\begin{bmatrix} 2 & -4 & 0 & 11\\ 0 & 0 & 3 & 6\\ 0 & 5 & -2 & 1\end{bmatrix}\) is not in row echelon form.
However, swapping rows 2 and 3 would put the matrix into row echelon form.
There are a few “limitations” with row echelon form worth noting:
The Good News: However, obtaining row echelon form can often be enough to give us several insights about a system.
In particular, we’ll see that row echelon form is enough to
If we want more than just these insights, then we’ll need to pursue an even further reduced form
We’ll perform additional row reduction in order to obtain reduced row echelon form.
When our goal is RREF, our target matrix is a matrix of the form
\[\left[\begin{array}{cccc|c} 1 & 0 & \cdots & 0 & b_1\\ 0 & 1 & \cdots & 0 & b_2\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 & 0 & \cdots & 1 & b_n\end{array}\right]\]
Example: The following matrices are all in reduced row echelon form. Again, the pivots are boxed up for convenience.
\[A = \left[\begin{array}{ccc|c} \boxed{~1~} & 0 & 0 & -5\\ 0 & \boxed{~1~} & 0 & 4\end{array}\right]~~~~~B = \left[\begin{array}{cc|c} \boxed{~1~} & 0 & 0\\ 0 & 0 & \boxed{~1~}\end{array}\right]\]
\[C = \left[\begin{array}{cccc|c} \boxed{~1~} & 0 & 0 & 0 & -3\\ 0 & \boxed{~1~} & 0 & 0 & 2\\ 0 & 0 & \boxed{~1~} & 0 & 0\\ 0 & 0 & 0 & \boxed{~1~} & 8\end{array}\right]~~~~~~D = \left[\begin{array}{ccccc|c} \boxed{~1~} & 2 & 0 & 1 & -8 & 2\\ 0 & 0 & \boxed{~1~} & -1 & 3 & -7\\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\]
From reduced row echelon form, we have an easier time reading solutions off of the augmented coefficient matrix.
Example: Revisit the augmented coefficient matrices in their reduced row echelon form. For each one, identify how many solutions the system has and use the matrices to describe the solution sets.
\[A = \left[\begin{array}{ccc|c} \boxed{~1~} & 0 & 0 & -5\\ 0 & \boxed{~1~} & 0 & 4\end{array}\right]~~~~~B = \left[\begin{array}{cc|c} \boxed{~1~} & 0 & 0\\ 0 & 0 & \boxed{~1~}\end{array}\right]\]
\[C = \left[\begin{array}{cccc|c} \boxed{~1~} & 0 & 0 & 0 & -3\\ 0 & \boxed{~1~} & 0 & 0 & 2\\ 0 & 0 & \boxed{~1~} & 0 & 0\\ 0 & 0 & 0 & \boxed{~1~} & 8\end{array}\right]~~~~~~D = \left[\begin{array}{ccccc|c} \boxed{~1~} & 2 & 0 & 1 & -8 & 2\\ 0 & 0 & \boxed{~1~} & -1 & 3 & -7\\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right]\]
Example: Use the following matrices in reduced row echelon form in order to describe the solution sets for the corresponding linear systems.
\[A = \left[\begin{array}{ccc|c} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 4 \end{array}\right]~~~~~B = \left[\begin{array}{ccc|c} 1 & 0 & 3 & 5 \\ 0 & 1 & -2 & 7 \\ 0 & 0 & 0 & 1 \end{array}\right]\]
\[C = \left[\begin{array}{cc|c} 1 & 0 & 3\\ 0 & 0 & 0\end{array}\right]\]
\[D = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 2\\ 0 & 1 & 0 & 0 & -3\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 5\end{array}\right]~~~~~E = \left[\begin{array}{ccc|c} 1 & 0 & -2 & 3\\ 0 & 0 & 0 & 1\end{array}\right]\]
Example: The following augmented coefficient matrices are in row echelon form. Prior to doing any additional row reduction, determine the number of solutions in the solution set for the corresponding system. Once you’ve done this, if solutions exist, reduce the matrix further in order to obtain its RREF form and describe the solutions to the corresponding system.
\[A = \left[\begin{array}{cc|c} -3 & 6 & -12\\ 0 & 1 & -1\end{array}\right]~~~~~B = \left[\begin{array}{ccc|c} 1 & 2 & -1 & 3\\ 0 & 2 & 8 & -4\\ 0 & 0 & 0 & -2\end{array}\right]~~~~~C = \left[\begin{array}{ccc|c} 3 & 9 & -6 & 15\\ 0 & 5 & 35 & -15\\ 0 & 0 & 1 & 3\end{array}\right]\]
\[D = \left[\begin{array}{cccc|c} 1 & 2 & 0 & 1 & 4\\ 0 & 1 & 3 & -1 & 2\\ 0 & 0 & 0 & -2 & -10\\ 0 & 0 & 0 & 0 & 0\end{array}\right]~~~~~E = \left[\begin{array}{ccc|c} 4 & -4 & 8 & 0\\ 0 & 1 & -3 & 1\\ 0 & 0 & 2 & -3\\ 0 & 0 & 0 & 7\end{array}\right]\]
Example: For the linear systems below, construct the corresponding augmented coefficient matrices, use row operations to reduce the matrices to their RREF form, and then describe the solution set for each system.
\[(A)~~\left\{\begin{array}{rcr} x + 2y -z & = & 4\\ 2x - y + 3z & = & 1\\ -3x + y +2z & = & -5\end{array}\right.~~~~~(B) \left\{\begin{array}{rcr} x_1 + x_2 + x_3 & = & 2\\ 2x_1 - x_2 + 3x_3 & = & 5\end{array}\right.\]
\[(C)~~\left\{\begin{array}{rcr}x_1 + x_2 + x_3 & = & 3\\ 2x_1 - x_2 + x_3 & = & 1\\ x_1 + 2x_2 - x_3 & = & 4\\ 3x_1 + x_2 + 2x_3 & = & 7\end{array}\right.~~~~~(D)~~\left\{\begin{array}{rcr}x_1 + 2x_2 - x_3 & = & 1\\ 2x_1 + 4x_2 -2x_3 & = & 2\\ -x_1 - 2x_2 + x_3 & = & -1\\ 3x_1 + 6x_2 - 3x_3 & = & 3\end{array}\right.\]
\[(E)~~\left\{\begin{array}{rcr} 2x_1 - 3x_2 & = & 6\\ 8x_1 - 12x_2 & = & 24\end{array}\right.\]
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\(\Huge{\text{Python for Computing}}\)