Gaussian elimination

Consider the following augmented matrix in which $\ast$ denotes an arbitrary number and $\blacksquare$ denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?

Consider the following augmented matrix in which $\ast$ denotes an arbitrary number and $\blacksquare$ denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?

Consider the following augmented matrix in which $\ast$ denotes an arbitrary number and $\blacksquare$ denotes a non-zero number. Determine whether the given augmented matrix is consistent. If consistent, is the solution unique?

Suppose a system of equations has fewer equations than variables. Will such a system necessarily be consistent? If so, explain why and if not, give an example which is not consistent.

If a system of equations has more equations than variables, can it have a solution? If so, give an example and if not, explain why not.

Find $h$ such that

is the augmented matrix of an inconsistent system.

Find $h$ such that

is the augmented matrix of a consistent system.

Find $h$ such that

is the augmented matrix of a consistent system.

Choose $h$ and $k$ such that the augmented matrix shown has each of the following:

1. one solution

2. no solution

3. infinitely many solutions

Choose $h$ and $k$ such that the augmented matrix shown has each of the following:

1. one solution

2. no solution

3. infinitely many solutions

Determine if the system is consistent. If so, is the solution unique?

Determine if the system is consistent. If so, is the solution unique?

Determine which matrices are in echelon form.

Find the general solution of the system whose augmented matrix is

Solve the system of equations $7x+14y+15z=22$, $2x+4y+3z=5$, and $3x+6y+10z=13$.

Solve the system of equations $3x-y+4z=6$, $y+8z=0$, and $-2x+y=-4$.

Solve the system of equations $9x-2y+4z=-17$, $13x-3y+6z=-25$, and $-2x-z=3$.

Solve the system of equations $65x+84y+16z=546$, $81x+105y+20z=682$, and $84x+110y+21z=713$.

Solve the system of equations $8x+2y+3z=-3$, $8x+3y+3z=-1$, and $4x+y+3z=-9$.

Suppose a system of equations has fewer equations than variables and you have found a solution to this system of equations. Is it possible that your solution is the only one? Explain.

Suppose a system of linear equations has an augmented matrix with 2 rows and 4 columns and the last column is a pivot column. Could the system of linear equations be consistent? Explain.

Suppose the coefficient matrix of a system of $n$ equations with $n$ variables has the property that every column is a pivot column. Does it follow that the system of equations must have a solution? If so, must the solution be unique? Explain.

Suppose there is a unique solution to a system of linear equations. What must be true of the pivot columns in the augmented matrix?

The steady state temperature, $u$, of a plate solves Laplace’s equation, $\Delta u=0$. One way to approximate the solution is to divide the plate into a square mesh and require the temperature at each node to equal the average of the temperature at the four adjacent nodes. In the following picture, the numbers represent the observed temperature at the indicated nodes. Find the temperature at the interior nodes…

Suppose $A$ is an $m\times n$-matrix. Explain why the rank of $A$ is always no larger than $\min(m,n)$.

State whether each of the following sets of data is possible for a system of equations. If possible, describe the solution set. That is, indicate whether there exists a unique solution, no solution or infinitely many solutions. Here, $A$ is the coefficient matrix, and $\left[A\mid B\right]$ denotes the augmented matrix of the system.

1. $A$ is a $5\times 6$-matrix, $\operatorname{rank}(A)=4$ and $\operatorname{rank}\left[A\mid B\right]=4$.

2. $A$ is a $3\times 4$-matrix, $\operatorname{rank}(A)=3$ and $\operatorname{rank}\left[A\mid B\right]=2$.

3. $A$ is a $4\times 2$-matrix, $\operatorname{rank}(A)=4$ and $\operatorname{rank}\left[A\mid B\right]=4$.

4. $A$…

Consider the system $-5x+2y-z=0$ and $-5x-2y-z=0$. Both equations equal zero and so $-5x+2y-z=-5x-2y-z$ which is equivalent to $y=0$. Does it follow that $x$ and $z$ can equal anything? Notice that when $x=1$, $z=-4$, and $y=0$ are plugged in to the equations, the equations do not equal $0$. Why?