Suppose that the augmented matrix for a system of linear...

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given (b) $\begin{array}{ll}10 & 1 \\ 1 & 3|3|\end{array}$ (c) $\left|\begin{array}{rrrrr|r}1 & 6 & 0 & 0 & 4 & -2 \mid \\ 0 & 0 & 1 & 0 & 3 & 1 \\ 0 & 0 & 0 & 1 & 5 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right|$ (d) $\left|\begin{array}{llll}0 & \mid 1 & 0 & 0|0| \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{array}\right|^{2}$

Suppose that the augmented matrix for a system of linear equations has been reduced by row operations to the given
(b) $\begin{array}{ll}10 & 1 \\ 1 & 3|3|\end{array}$
(c) $\left|\begin{array}{rrrrr|r}1 & 6 & 0 & 0 & 4 & -2 \mid \\ 0 & 0 & 1 & 0 & 3 & 1 \\ 0 & 0 & 0 & 1 & 5 & 2 \\ 0 & 0 & 0 & 0 & 0 & 0\end{array}\right|$
(d) $\left|\begin{array}{llll}0 & \mid 1 & 0 & 0|0| \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1\end{array}\right|^{2}$

Key Concepts

Pivot Positions and Leading Variables

Pivot positions are the locations in a matrix that contain the first nonzero entries in each row when in echelon form. These positions identify the leading variables in the system and are central to understanding the rank of the matrix, which in turn indicates the number of independent constraints acting on the variables.

Reduced Row Echelon Form

Reduced row echelon form further refines the row echelon form by ensuring that every pivot is 1 and that all other entries in the pivot columns are zeros. This form provides a clear and direct view of the relationships between variables, making it easier to deduce the unique or parametric solutions for the system.

Consistency of Systems

The consistency of a system of linear equations refers to whether the system has at least one solution. A consistent system may have a unique solution or infinitely many solutions, while an inconsistent system has no solution. This is often determined after transforming the augmented matrix via row operations, where a row with all zeros in the coefficient part but a nonzero constant indicates an inconsistency.

Row Operations

Row operations, including row swapping, scaling, and row addition, are used to manipulate the rows of a matrix without changing the solution set of the system of linear equations. They are essential techniques for transforming an augmented matrix into row echelon or reduced row echelon form.

Augmented Matrix

An augmented matrix represents a system of linear equations by combining the coefficient matrix with the constants into a single matrix. This approach provides a streamlined method to apply systematic procedures, such as row operations, in order to solve or analyze the system.

Row Echelon Form

Row echelon form is a simplified version of the augmented matrix in which all nonzero rows are above any rows of all zeros, and each leading coefficient (pivot) in a row is to the right of the pivot in the row above. This form assists in determining the presence and nature of solutions by highlighting the basic structure of the system.

Transcript

00:01 The question asked to solve this reduced echelon form augmented matrix.

00:08 So as you already may know, a augmented matrix has variables in each column.

00:18 It has, well, you can name this any variable, but let's say that this first column contains x1 variables.

00:25 This second column contains x2 variables.

00:28 And this third column contains x3 variable, and the very last column in an augmented matrix would be the solution to this.

00:38 So we can rewrite this as a system of equations.

00:43 So if we look at the first row, we can say that x1 plus, well, 0x2 plus 0x3 equals negative 3.

00:52 And in the second row, we can say that x2, well, 0x1 plus x2 plus 0x3 equals 0.

01:02 And for at the last row, we can see that 0x1 plus 0x2 plus x3 equals 7.

01:12 So the solutions for this matrix would be negative x1 equals negative 3, x2 equals 0, and x3 equals 7.

01:22 So this matrix has a unique solution.

01:28 Looking at letter b, we can apply the same strategy.

01:33 So let's say that this column is x1, the second one is x2, x3, x4, and the last row is is b or our solution column.

01:45 Rewriding this as a system of equations, we can write x1 plus 0 x2, 0x3 minus 0, minus 7 x4 equals 8.

01:57 The second rule tells us that there is 0x1 plus x2 plus 0x3 plus 0x4 equals 2.

02:07 And the last column tells us that there is 0x1 plus 0x2 plus x3 plus x4 equals b.

02:18 And you notice that there, we only have three systems of equations, but there are actually for variables.

02:30 And for this particular matrix, x4, in this case, would be a free variable because the leading entries are only, only exist in the x1 column, x2 column, x3 column...

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