1.2 Exercises

20 points · 4 extra credit

Row Reduction and Echelon Forms

1. 1. (2 pts) Row reduce the matrix to reduced echelon form and list its pivot positions and pivot columns:

   $A = \begin{bmatrix} 1 & 2 & 1 & 4 \\ 2 & 4 & 3 & 9 \\ 3 & 6 & 4 & 13 \end{bmatrix}.$

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2. 2. (3 pts) Solve the following system, giving a parametric description of its general solution (solve for the basic variables in terms of the free variable):

   $\begin{aligned} x_1 + 2x_2 - x_3 &= 3 \\ 2x_1 + 5x_2 - x_3 &= 7. \end{aligned}$

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3. 3. (3 pts) For the coefficient matrix $\begin{bmatrix} 1 & -2 \\ 2 & h \end{bmatrix}$, find the value of $h$ for which it fails to have a pivot in every column. For that value, determine whether the system with augmented matrix $\left[\begin{array}{cc|c} 1 & -2 & 3 \\ 2 & h & 5 \end{array}\right]$ is consistent.

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4. 4. (3 pts) Determine whether each matrix is in (i) echelon form, (ii) reduced echelon form. If it is not in echelon form, state which condition fails.

   (a) $\begin{bmatrix} 1 & 2 & 0 \\ 0 & 0 & 3 \\ 0 & 1 & 0 \end{bmatrix}$

   (b) $\begin{bmatrix} 1 & 0 & 3 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \end{bmatrix}$

   (c) $\begin{bmatrix} 2 & 1 & 0 \\ 0 & 0 & 5 \\ 0 & 0 & 0 \end{bmatrix}$

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5. 5. (3 pts) State whether each of the following is true or false, and justify each answer.

   (a) The reduced echelon form of a matrix is unique.

   (b) Every matrix has a unique echelon form.

   (c) If a coefficient matrix has a pivot position in every row, then the corresponding system is consistent for every right-hand side.

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6. 6. (3 pts) Explain why a system of three equations in four unknowns can never have a unique solution. What are the only possibilities for its solution set?

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7. 7. (3 pts) The augmented matrix of a linear system has been reduced to

   $\left[\begin{array}{cccc|c} 1 & -2 & 0 & 3 & 5 \\ 0 & 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right].$

   Identify the basic and free variables, and give a parametric description of the general solution (each basic variable in terms of the free variables).

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8. 8. (4 pts ••) Prove that a linear system is consistent if and only if the rightmost column of its augmented matrix is not a pivot column — equivalently, no row of an echelon form of the augmented matrix has the form $[\,0 \;\; \cdots \;\; 0 \mid b\,]$ with $b \neq 0$.

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