Question 1

An m × n zero matrix serves as an m × n .

Accepted Answer

additive identity

Question 2

Accepted Answer

The answer of ...

Question 3

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place. $$
2.7 x-0.2 y-5.0 z=2.6
$$
$ 5.6 x+4.6 y-0.5 z=-4.1 $
$ 3.4 x-1.3 y+1.6 z=10.5 $
A) $ (1.7,-2.9,0.5) $
B) $ (8.6,-14.6,2.6) $
C) $ (6.9,-11.7,2.1) $
D) $ (3.4,-5.9,1.1) $

Accepted Answer

Using a graphing calculator to solve the system of equations and rounding the solution to one decimal place yields the solution $ (1.7,-2.9,0.5) $.

Question 4

Provide an appropriate response
-Which choice best describes the following matrix? $$
\left[\begin{array}{l}
24 \
25 \
20 \
17 \
18
\end{array}\right]
$$
A) Column matrix
B) Row matrix
C) Square matrix
D) $ 1 \times 5 $ matrix

Accepted Answer

This matrix has 5 rows and 1 column, which defines it as a column matrix.

Question 5

Perform the indicated operation, where possible.
-$$
\left[\begin{array}{r}
3 \
-1 \
-4
\end{array}\right]+\left[\begin{array}{r}
-5 \
2 \
8
\end{array}\right]
$$
A) $ \left[\begin{array}{l}2 \ 4 \ 5\end{array}\right] $
B) $ \left[\begin{array}{lll}-2 & 1 & 4\end{array}\right] $
C) $ \left[\begin{array}{r}-2 \ 1 \ 4\end{array}\right] $
D) $ \left[\begin{array}{rr}3 & -5 \ -1 & 2 \ -4 & 8\end{array}\right] $

Accepted Answer

To add two matrices, add their corresponding elements. Thus, $3 + (-5) = -2$, $-1 + 2 = 1$, and $-4 + 8 = 4$, resulting in $\left[\begin{array}{r}-2 \ 1 \ 4\end{array}\right]$.

Question 6

Use the Gauss-Jordan method to solve the system of equations.$$
\begin{array}{l}
3 x-2 y=-3 \
9 x-6 y=-9
\end{array}
$$
A) $ (1,3) $
B) $ \left(-1-\frac{2}{3} y, y\right) $
C) $ \left(-1+\frac{2}{3} y, y\right) $
D) No solution

Accepted Answer

The answer of Use the Gauss-Jordan method to solve the...

Question 7

Solve the problem.
-Janet is planning to visit Arizona, New Mexico, and California on a 20-day vacation. If she plans to spend as much time in New Mexico as she does in the other two states combined, how can she allot her time in the three states? (Let $ x $ denote the number of days in Arizona, $ y $ the number of days in New Mexico, and $ z $ the number of days in California. Let $ z $ be the parameter.)
A) $ x=10, y=10-z, 0 \leq z \leq 10 $
B) $ x=10, y=z-10,0 \leq z \leq 10 $
C) $ x=z-10, y=10,0 \leq z \leq 10 $
D) $ x=10-z, y=10,0 \leq z \leq 10 $

Accepted Answer

Janet spends as much time in New Mexico as in Arizona and California combined, so $ y = x + z $. The total time is 20 days, so $ x + y + z = 20 $. Substituting $ y = x + z $ gives $ x + (x + z) + z = 20 $, simplifying to $ 2x + 2z = 20 $, or $ x + z = 10 $. Thus, $ x = 10 - z $ and $ y = 10 $.

Question 8

Does a matrix with a column of all zeros have an inverse? Why or why not?

Accepted Answer

The answer of Does a matrix with a column of...

Question 9

Use the Gauss-Jordan method to solve the system of equations.
-$$
\begin{array}{l}
5 x-y+z=8 \
7 x+y+z=6 \
12 x+2 z=14
\end{array}
$$
A) $ \left(\frac{-z+7}{6}, \frac{z-13}{6}, z\right) $
B) $ \left(\frac{z+7}{6}, \frac{-z-13}{6}, z\right) $
C) $ \left(\frac{-z+7}{6}, \frac{z+13}{6}, z\right) $
D) $ \left(\frac{z-7}{6}, \frac{-z-13}{6}, z\right) $

Accepted Answer

To solve the system using the Gauss-Jordan method, we first write the augmented matrix of the system and then apply row operations to get it in reduced row echelon form (RREF). The system of equations is:$$\begin{align*}5x - y + z &= 8 \7x + y + z &= 6 \12x + 0y + 2z &= 14\end{align*}$$The augmented matrix is:$$\begin{bmatrix}5 & -1 & 1 & | & 8 \7 & 1 & 1 & | & 6 \12 & 0 & 2 & | & 14\end{bmatrix}$$Applying row operations to simplify this matrix into RREF, we aim to get leading ones in each row and zeros everywhere else in the column of those leading ones, except in the last column which represents the constants from the right-hand side of the equations.After performing the necessary row operations (which are too lengthy to detail in this brief explanation but involve making the leading coefficient of each row a 1, eliminating other numbers in that column to 0, and making sure each leading 1 moves to the right as we move down the rows), we find that the system of equations has one free variable (let's choose $z$ as the free variable since it's common to express the solution in terms of the last variable when possible).The solution in terms of $z$ matches option A: $ \left(\frac{-z+7}{6}, \frac{z-13}{6}, z\right) $. This indicates that $x$ and $y$ can be expressed in terms of $z$, and their specific relationships to $z$ are given in option A, which is derived from the final form of the augmented matrix after applying Gauss-Jordan elimination.

Question 10

The property does not apply to matrix multiplication.

Accepted Answer

The answer of The property does not apply to matrix...

Question 11

Accepted Answer

The answer of ...

Question 12

If a variable is expressed in terms of another variable, what is the other variable called?

Accepted Answer

The answer of If a variable is expressed in terms...

Question 13

How many solutions are there to a dependent system?

Accepted Answer

The answer of How many solutions are there to a...

Question 14

Describe the proper form for the system to be in before the Gauss-Jordan method can be  used.

Accepted Answer

The answer of Describe the proper form for the system...

Question 15

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place.$$
\begin{array}{l}
1.5 x-0.4 y+1.6 z=2.4 \
4.5 x-7.0 y-0.4 z=-4.0 \
3.8 x+2.4 y+2.0 z=8.6
\end{array}
$$
A) $ (0.8,0.9,0.6) $
B) $ (1.1,1.2,0.8) $
C) $ (0.5,0.6,0.4) $
D) $ (0.3,0.3,0.2) $

Accepted Answer

The answer of Use a graphing calculator to solve the...

Question 16

Use the echelon method to solve the system.$$
\begin{array}{l}
\frac{x}{5}+\frac{y}{5}=1 \
\frac{x}{5}-\frac{y}{5}=-\frac{7}{5}
\end{array}
$$
A) $ (1,7) $
B) $ (-2,7) $
C) $ (-1,6) $
D) No solution

Accepted Answer

The answer of Use the echelon method to solve the...

Question 17

Can non-square matrices have inverses? Why or why not?

Accepted Answer

The answer of Can non-square matrices have inverses? Why or...

Question 18

Use the Gauss-Jordan method to solve the system of equations.$$
\begin{array}{l}
-2 x-8 y=6 \
-4 x-16 y=2
\end{array}
$$
A) $ (2,2) $
B) $ \left(-\frac{4}{3} x-\frac{1}{3} y, y\right) $
C) $ (6,2) $
D) No solution

Accepted Answer

The given system of equations is:$$\begin{array}{l}-2x - 8y = 6 \-4x - 16y = 2\end{array}$$If we divide the second equation by 2, we get:$$-2x - 8y = 1$$Comparing this with the first equation:$$-2x - 8y = 6$$We see that the two equations are inconsistent (they cannot both be true for the same values of $x$ and $y$) because they have the same left-hand side but different right-hand sides. This means the system of equations has no solution.

Question 19

Use the Gauss-Jordan method to solve the system of equations.$ 8 w+8 x-6 y-2 z=-30 $
$ 7 w+6 x-9 y-2 z=-38 $
$ 8 w+8 x+7 y+3 z=17 $
$ -6 w-2 x+8 y+8 z=18 $

A)  (-1,4,-5,2)  
B) (2,3,-4,-2)  
C) (2,-3,4,-1)  
D) No solution

Accepted Answer

The answer of Use the Gauss-Jordan method to solve the...

Question 20

Is this a square matrix? Why or why not?

Accepted Answer

The answer of Is this a square matrix? Why or...

Quiz 2: Systems of Linear Equations and Matrices

An m × n zero matrix serves as an m × n .

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place. $2.7 x-0.2 y-5.0 z=2.6$ $5.6 x+4.6 y-0.5 z=-4.1$ $3.4 x-1.3 y+1.6 z=10.5$

Provide an appropriate response -Which choice best describes the following matrix? $\left[\begin{array}{l}24 \\25 \\20 \\17 \\18\end{array}\right]$

Perform the indicated operation, where possible. - $\left[\begin{array}{r}3 \\-1 \\-4\end{array}\right]+\left[\begin{array}{r}-5 \\2 \\8\end{array}\right]$

Use the Gauss-Jordan method to solve the system of equations. $\begin{array}{l}3 x-2 y=-3 \\9 x-6 y=-9\end{array}$

Does a matrix with a column of all zeros have an inverse? Why or why not?

Use the Gauss-Jordan method to solve the system of equations. - $\begin{array}{l}5 x-y+z=8 \\7 x+y+z=6 \\12 x+2 z=14\end{array}$

The property does not apply to matrix multiplication.

If a variable is expressed in terms of another variable, what is the other variable called?

How many solutions are there to a dependent system?

Describe the proper form for the system to be in before the Gauss-Jordan method can be used.

Use a graphing calculator to solve the system of equations. Round your solution to one decimal place. $\begin{array}{l}1.5 x-0.4 y+1.6 z=2.4 \\4.5 x-7.0 y-0.4 z=-4.0 \\3.8 x+2.4 y+2.0 z=8.6\end{array}$

Use the echelon method to solve the system. $\begin{array}{l}\frac{x}{5}+\frac{y}{5}=1 \\\frac{x}{5}-\frac{y}{5}=-\frac{7}{5}\end{array}$

Can non-square matrices have inverses? Why or why not?

Use the Gauss-Jordan method to solve the system of equations. $\begin{array}{l}-2 x-8 y=6 \\-4 x-16 y=2\end{array}$

Use the Gauss-Jordan method to solve the system of equations. $8 w+8 x-6 y-2 z=-30$ $7 w+6 x-9 y-2 z=-38$ $8 w+8 x+7 y+3 z=17$ $-6 w-2 x+8 y+8 z=18$

Is this a square matrix? Why or why not?

Quiz 2: Systems of Linear Equations and Matrices

An m × n zero matrix serves as an m × n .

Provide an appropriate response -Which choice best describes the following matrix? [2425201718]\left[\begin{array}{l}24 \\25 \\20 \\17 \\18\end{array}\right]​2425201718​​

Perform the indicated operation, where possible. - [3−1−4]+[−528]\left[\begin{array}{r}3 \\-1 \\-4\end{array}\right]+\left[\begin{array}{r}-5 \\2 \\8\end{array}\right]​3−1−4​​+​−528​​

Use the Gauss-Jordan method to solve the system of equations. 3x−2y=−39x−6y=−9\begin{array}{l}3 x-2 y=-3 \\9 x-6 y=-9\end{array}3x−2y=−39x−6y=−9​

Does a matrix with a column of all zeros have an inverse? Why or why not?

Use the Gauss-Jordan method to solve the system of equations. - 5x−y+z=87x+y+z=612x+2z=14\begin{array}{l}5 x-y+z=8 \\7 x+y+z=6 \\12 x+2 z=14\end{array}5x−y+z=87x+y+z=612x+2z=14​

The property does not apply to matrix multiplication.

If a variable is expressed in terms of another variable, what is the other variable called?

How many solutions are there to a dependent system?

Describe the proper form for the system to be in before the Gauss-Jordan method can be used.

Use the echelon method to solve the system. x5+y5=1x5−y5=−75\begin{array}{l}\frac{x}{5}+\frac{y}{5}=1 \\\frac{x}{5}-\frac{y}{5}=-\frac{7}{5}\end{array}5x​+5y​=15x​−5y​=−57​​

Can non-square matrices have inverses? Why or why not?

Use the Gauss-Jordan method to solve the system of equations. −2x−8y=6−4x−16y=2\begin{array}{l}-2 x-8 y=6 \\-4 x-16 y=2\end{array}−2x−8y=6−4x−16y=2​

Use the Gauss-Jordan method to solve the system of equations. 8w+8x−6y−2z=−30 8 w+8 x-6 y-2 z=-30 8w+8x−6y−2z=−30 7w+6x−9y−2z=−38 7 w+6 x-9 y-2 z=-38 7w+6x−9y−2z=−38 8w+8x+7y+3z=17 8 w+8 x+7 y+3 z=17 8w+8x+7y+3z=17 −6w−2x+8y+8z=18 -6 w-2 x+8 y+8 z=18 −6w−2x+8y+8z=18

Is this a square matrix? Why or why not?

Provide an appropriate response -Which choice best describes the following matrix? $\left[\begin{array}{l}24 \\25 \\20 \\17 \\18\end{array}\right]$

Perform the indicated operation, where possible. - $\left[\begin{array}{r}3 \\-1 \\-4\end{array}\right]+\left[\begin{array}{r}-5 \\2 \\8\end{array}\right]$

Use the Gauss-Jordan method to solve the system of equations. $\begin{array}{l}3 x-2 y=-3 \\9 x-6 y=-9\end{array}$

Use the Gauss-Jordan method to solve the system of equations. - $\begin{array}{l}5 x-y+z=8 \\7 x+y+z=6 \\12 x+2 z=14\end{array}$

Use the echelon method to solve the system. $\begin{array}{l}\frac{x}{5}+\frac{y}{5}=1 \\\frac{x}{5}-\frac{y}{5}=-\frac{7}{5}\end{array}$

Use the Gauss-Jordan method to solve the system of equations. $\begin{array}{l}-2 x-8 y=6 \\-4 x-16 y=2\end{array}$

Use the Gauss-Jordan method to solve the system of equations. $8 w+8 x-6 y-2 z=-30$ $7 w+6 x-9 y-2 z=-38$ $8 w+8 x+7 y+3 z=17$ $-6 w-2 x+8 y+8 z=18$