Math 8, Solutions to Exercises Due May 5, 2000

Exercise 1   For each of the following systems of linear equations, sketch on the same graph the lines described by each equation, and determine whether the system has no solutions, one solution, or infinitely many solutions.

$$x + y = 1$$

$$3x + 3y = -3$$

These are two parallel lines; there are no solutions.

$$x + 2y = 2$$

$$2x + 4y = 4$$

These are the same line; there are infinitely many solutions.

$$3x - y = 3$$

$$x - y = 3$$

These two lines intersect at the point $(x,y) = (0,-3)$; there is one solution.

$$x + y = 1$$

$$x - y = 1$$

$$2y-x=2$$

These three lines form the sides of a triangle; there are no solutions.

$$x + y = 1$$

$$x - y = -1$$

$$2y-x=2$$

These three lines intersect at the point $(x,y) = (0,1)$; there is one solution.

$$x + y = 1$$

$$x - y = 1$$

$$y - x =1$$

Two of these lines are parallel; there are no solutions.

Exercise 2   Determine which of the following matrices are in row echelon form. For those which are not, give a sequence of elementary row operations that will put them in row echelon form.

$$\left(\begin{array}{cccc} 1 & 0 & 3 & 2 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

NO: Add $-2$ times row 3 to row 1.

$$\left(\begin{array}{ccc} 1 & 2 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$$

NO: Add $-2$ times row 2 to row 1.

$$\left(\begin{array}{ccc} 1 & -1 & 3 \\ 0 & 1 & 2 \end{array}\right)$$

NO: Add row 2 to row 1.

$$\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right)$$

NO: Interchange rows 1 and 2.

$$\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)$$

YES.

$$\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 2 \end{array}\right)$$

NO: Multiply row 2 by $-1$; multiply row 3 by $$\frac{1}{2}$$.

Exercise 3   Solve the system of linear equations

$$w + x + y + z = 2$$

$$w - x - y = 0$$

$$3w + y + 2z = 5$$

$$6w + x + 2y +4z = 9$$

and give the solution in parametric form.

One solution to this system of equations is $w=1$, $x=-1$, $y=2$, and $z=0$. What choice of parameters in your solution gives this particular solution?

$$w = 1-\frac{1}{2}t$$

$$x = -1$$

$$y = 2-\frac{1}{2}t$$

$$z = t$$

The given solution is obtained by setting $t=0$.

Exercise 4   Solve the following systems of linear equations by row-reducing their augmented matrices.

$$x + y = 1$$

$$x - y = 3$$

$$\left(\begin{matrix} 1 & 1 & \vdots & 1 \cr 1 & -1 & \vdots & 3 \... ...gin{matrix} 1 & 0 & \vdots & 1 \cr 0 & 1 & \vdots & -1 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} x \cr y \cr \end{matrix}\right) = \left(\begin{matrix} 2 \cr -1 \cr \end{matrix}\right)$$

$$x + y - z = 1$$

$$2x + 3y + z = 6$$

$$x - y + 2z = 2$$

$$\left(\begin{matrix} 1 & 1 & -1 &\vdots & 1 \cr 2 & 3 & 1 &\vdots... ... 1 \cr 0 & 1 & 0 &\vdots & 1 \cr 0 & 0 & 1 &\vdots & 1 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} x \cr y \cr z \cr \end{matrix}\right) = \left(\begin{matrix} 1 \cr 1 \cr 1 \cr \end{matrix}\right)$$

$$y + z = 0$$

$$2x - y - z = 4$$

$$2x + 2y + 2z = 2$$

$$\left(\begin{matrix} 0 & 1 & 1 &\vdots & 0 \cr 2 & -1 & -1 &\vdot... ... 0 \cr 0 & 1 & 1 &\vdots & 0 \cr 0 & 0 & 0 &\vdots & 1 \cr \end{matrix}\right)$$

There are no solutions.

$$v + w - x = 1$$

$$2v - w = 1$$

$$3v - x = 2$$

$$\left(\begin{matrix} 1 & 1 & -1 &\vdots & 1 \cr 2 & -1 & 0 &\vdot... ...{2}{3} &\vdots & \frac{1}{3} \cr 0 & 0 & 0 &\vdots & 0 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} v \cr w \cr x \cr \end{matrix}\right) = \lef... ...3}t + \frac{2}{3} \cr \frac{2}{3}t + \frac{1}{3} \cr t \cr \end{matrix}\right)$$

Exercise 5   Explain why a system of two linear equations in three variables may have no solutions or infinitely many solutions, but will never have exactly one solution.

Which outcomes (no solutions, one solution, infinitely many solutions) can possibly occur in systems of four linear equations in three variables?

The solutions to two linear equations in three variables represent the points in the intersection of two planes in three-dimensional space. Two planes can intersect in a line or a plane (in which case the system has infinitely many solutions), or they can be parallel and not intersect at all (in which case the system has no solutions), but they can never intersect in exactly one point.

All three outcomes are possibilties for systems of four linear equations in three variables.

Math 8, Solutions to Exercises Due May 8, 2000

Exercise 6   Rewrite the systems of linear equations of Exercise 4 (in the last handout) as matrix equations.

$$\left(\begin{matrix} 1 &1 \cr 1 &-1 \cr \end{matrix}\right) \left... ...y \cr \end{matrix}\right)= \left(\begin{matrix} 1 \cr 3 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} 1 &1 &-1 \cr 2 &3 &1 \cr 1 &-1 &2 \cr \end{m... ...\end{matrix}\right)= \left(\begin{matrix} 1 \cr 6 \cr 2 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} 0 &1 & 1\cr 2 &-1 & -1\cr 2 &2 & 2\cr \end{m... ...\end{matrix}\right)= \left(\begin{matrix} 0 \cr 4 \cr 2 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} 1 & 1& -1\cr 2 &-1 &0 \cr 3 &0 &-1 \cr \end{... ...\end{matrix}\right)= \left(\begin{matrix} 1 \cr 1 \cr 2 \cr \end{matrix}\right)$$

Exercise 7   Rewrite the following matrix equations as systems of linear equations.

$$\left(\begin{array}{ccc} 1 & 1 & 1 \\ 3 & 1 & -1 \end{array}\rig... ... y \\ z \end{array}\right) = \left(\begin{array}{c} 4 \\ 2 \end{array}\right)$$

$$x+y+x=4$$

$$3x+y-z=2$$

$$\left(\begin{array}{cc} 3 & 1 \\ 0 & 1 \\ 9 & 8 \\ -1 & 1 \en... ...{array}\right) = \left(\begin{array}{c} -1 \\ 1 \\ 3 \\ 2 \end{array}\right)$$

$$3x+y=-1$$

$$y=1$$

$$9x+8y=3$$

$$-x+y=2$$

$$\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \e... ... z \end{array}\right) = \left(\begin{array}{c} 3 \\ 1 \\ 4 \end{array}\right)$$

$$x=3$$

$$y=1$$

$$z=4$$

Exercise 8   Carry out the following matrix multiplications, or explain why they cannot be carried out.

$$\left(\begin{array}{cc} 3 & 2 \\ 2 & 1 \end{array}\right) \left... ... \\ -1 \end{array}\right) =\left(\begin{matrix} 1\cr 1\cr \end{matrix}\right)$$

$$\left(\begin{array}{ccc} 1 & 2 & 4 \\ 0 & 1 & 3 \end{array}\righ... ...2 \\ 1 \end{array}\right) =\left(\begin{matrix} 7\cr 5\cr \end{matrix}\right)$$

$$\left(\begin{array}{ccc} 3 & 1 & 2 \\ 1 & -1 & 4 \end{array}\right) \left(\begin{array}{c} 2 \\ 9 \end{array}\right)$$

This cannot be done; the number of entries in the rows of the left-hand matrix is different from the number of entries in the columns of the right-hand matrix.

$$\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right) \left(... ... x \\ y \end{array}\right) =\left(\begin{matrix} x\cr y\cr \end{matrix}\right)$$

Exercise 9   Solve the following matrix equations using row-reduction.

$$\left(\begin{array}{cc} 1 & 3 \\ -1 & 4 \end{array}\right) \lef... ...x \\ y \end{array}\right) = \left(\begin{array}{c} 4 \\ 3 \end{array}\right)$$

$$\left(\begin{matrix} 1 &3& \vdots & 4\cr -1 & 4 & \vdots& 3 \cr \... ...\begin{matrix} 0 &1 & \vdots & 1 \cr 1 &0 & \vdots & 1 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} x \cr y \cr \end{matrix}\right)= \left(\begin{matrix} 1 \cr 1 \cr \end{matrix}\right)$$

$$\left(\begin{array}{ccc} 2 & 1 & -1 \\ 4 & 2 & 8 \end{array}\rig... ...w \\ x \end{array}\right) = \left(\begin{array}{c} 6 \\ 2 \end{array}\right)$$

$$\left(\begin{matrix} 2 &1 &-1 & \vdots & 6\cr 4 &2 &8 & \vdots& 2... ...2} & 0& \vdots & \frac{5}{2} \cr 0 &0 &1 & \vdots & -1 \cr \end{matrix}\right)$$

$$\left(\begin{matrix} v \cr w \cr x \cr \end{matrix}\right)= \left... ...ght) + \left(\begin{matrix} \frac{5}{2} \cr 0 \cr -1 \cr \end{matrix}\right)$$

Exercise 10   What does it say about the set of solutions to a system of linear equations if its augmented matrix, when put into row-reduced form:

(a.) Has a row whose leading entry is in the last column (the column corresponding to the constant terms)?

The equation corresponding to that row is $0=1$; there are no solutions to the system.

(b.) Has all zeroes in the last column?

The system is homogeneous; it has at least one solution, namely, every variable has value zero.

(c.) Has a column, other than the last column, in which no row has a leading entry?

There is a variable that is assigned a parameter; the system has infinitely many solutions (unless there is also a row with leading entry in the last column, in which case it has no solutions.)

(d.) Has rows with leading entries in every column except the last one?

The system has a unique solution.

Exercise 11   Write down the associated homogeneous matrix equations for the matrix equations in exercise 9. Now write down the complete solution to each of these homogeneous matrix equations.

$$\left(\begin{array}{cc} 1 & 3 \\ -1 & 4 \end{array}\right) \lef... ...x \\ y \end{array}\right) = \left(\begin{array}{c} 0 \\ 0 \end{array}\right)$$

$$\left(\begin{matrix} x \cr y \cr \end{matrix}\right)= \left(\begin{matrix} 0 \cr 0 \cr \end{matrix}\right)$$

$$\left(\begin{array}{ccc} 2 & 1 & -1 \\ 4 & 2 & 8 \end{array}\rig... ...w \\ x \end{array}\right) = \left(\begin{array}{c} 0 \\ 0 \end{array}\right)$$

$$\left(\begin{matrix} v \cr w \cr x \cr \end{matrix}\right)= t \left(\begin{matrix} -\frac{1}{2} \cr 1 \cr 0\cr \end{matrix}\right)$$

Exercise 12   The matrix equation

$$\left(\begin{array}{ccc} 2 & 1 & -1 \\ 4 & 2 & 8 \end{array}\rig... ...w \\ x \end{array}\right) = \left(\begin{array}{c} 1\\ 26 \end{array}\right)$$

has one solution given by

$$\left(\begin{array}{c} v \\ w \\ x \end{array}\right) = \left(\begin{matrix}2 \cr 1 \cr 2 \cr \end{matrix}\right).$$

Give the complete solution to this matrix equation.

$$\left(\begin{matrix} v \cr w \cr x \cr \end{matrix}\right)= t \le... ...end{matrix}\right) + \left(\begin{matrix}2 \cr 1 \cr 2 \cr \end{matrix}\right)$$

Math 8, Solutions to Exercises Due May 10, 2000

Exercise 13   For each of the following matrix products, state whether the product is defined, and if it is, what the dimensions of the product will be. You do not have to do the multiplication.

$$\left(\begin{array}{cc} 3 & 2 \\ 1 & 4 \end{array}\right) \left(\begin{array}{ccc} 5 & -1 & 0 \\ 2 & 5 & 1 \end{array}\right)$$

$$\left(\begin{array}{ccc} 5 & -1 & 0 \\ 2 & 5 & 1 \end{array}\right) \left(\begin{array}{cc} 3 & 2 \\ 1 & 4 \\ 5 & -2 \end{array}\right)$$

$$\left(\begin{array}{cc} 2 & 4 \\ 1 & 3 \\ 5 & 1 \end{array}\right) \left(\begin{array}{ccc} 1 & 0 & 2 \\ 5 & 3 & 6 \end{array}\right)$$

$$\left(\begin{array}{ccc} 9 & 5 & 2 \\ 4 & -4 & 1 \end{array}\rig... ...ft(\begin{array}{ccc} 5 & 6 & 5 \\ 3 & 6 & 1 \\ 0 & 0 & 0 \end{array}\right)$$

$$\left(\begin{array}{cc} a & b \\ c & d \\ \end{array}\right) \left(\begin{array}{c} x \\ y \end{array}\right)$$

Exercise 14   Carry out the following matrix multiplications.

$$\left(\begin{array}{cc} 1 & 2 \\ -2 & -4 \end{array}\right) \left(\begin{array}{cc} 3 & -2 \\ 1 & 1 \end{array}\right)$$

$$\left(\begin{array}{cc} 1 & 2 \\ -2 & -4 \end{array}\right) \left(\begin{array}{cc} 9 & 4 \\ -2 & -2 \end{array}\right)$$

$$\left(\begin{array}{cc} 3 & 1 \\ 5 & 4 \end{array}\right) \left(\begin{array}{cc} 8 & 9 \\ 1 & 2 \end{array}\right)$$

$$\left(\begin{array}{cc} 8 & 9 \\ 1 & 2 \end{array}\right) \left(\begin{array}{cc} 3 & 1 \\ 5 & 4 \end{array}\right)$$

$$\left(\begin{array}{ccc} 3 & 2 & 1 \\ 4 & 9 & 8 \end{array}\righ... ...ray}{ccc} x & y & z \\ x^2 & y^2 & z^2 \\ x^3 & y^3 & z^3 \end{array}\right)$$

Exercise 15   Find inverses for all of the following matrices that are invertible.

$$\left(\begin{array}{ccc} 1 & 2 & 1 \\ 2 & 3 & 3 \\ 4 & 9 & 6 \end{array}\right)$$

$$\left(\begin{array}{ccc} 1 & 2 & 1 \\ 2 & 3 & 3 \\ 4 & 7 & 5 \end{array}\right)$$

$$\left(\begin{array}{ccc} 9 & 3 & -3 \\ 0 & -2 & 1 \\ -2 & 1 & 1 \end{array}\right)$$

$$\left(\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 \end{array}\right)$$

$$\left(\begin{array}{cccc} 3 & 5 & 2 & 7 \\ 3 & 5 & 2 & 7 \\ 11 & 4 & 3 & 15 \\ -4 & 5 & 19 & 3 \end{array}\right)$$

Exercise 16   Solve each of these systems of simultaneous linear equations. (Hint: Convert to a matrix equation $AX=B$, and check for the matrix $A$ in the preceding exercise.)

$$x + 2y + z = 3$$

$$2x + 3y + 3z = 3$$

$$4x + 9y + 6z = 3$$

$$$$

$$x + 2y + z = 1$$

$$2x + 3y + 3z = 3$$

$$4x + 9y + 6z = 9$$

$$$$

$$x + 2y + z = 6$$

$$2x + 3y + 3z = 12$$

$$4x + 9y + 6z = -3$$

$$$$

$$x + 2y + z = 0$$

$$2x + 3y + 3z = 0$$

$$4x + 7y + 5z = 0$$

Exercise 17   Does the system of simultaneous linear equations

$$x + 2y + z = 0$$

$$3x + 4z = 0$$

$$8x + 4y + 10z = 0$$

have no solutions, one solution, or infinitely many solutions? How do you know?

Exercise 18   Put the matrix

$$\left(\begin{array}{cccc} 1 & 2 & 2 & 1 \cr 0 & 2 & 0 & 0 \cr -1 & -4 & -2 & -1 \cr 1 & 4 & 4 & 1 \cr \end{array}\right)$$

into row echelon form. Then say whether or not the determinant of this matrix is zero.

Exercise 19   Find the determinant of the matrix

$$\left(\begin{array}{cccc} 2 &1 & 4& 8 \cr 0 & 2 & 5 &19 \cr 0 & 0 & 3 & -1 \cr 2 & 1 & 4 & 0 \cr \end{array}\right).$$