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Finite Mathematics : Systems of Linear Equations: Matrices

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Example Questions

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Example Question #4 : Finite Mathematics

Find the value of when,

$6\times 7\equiv n\ (\text{mod } 5)$ .

Possible Answers:

n=42

n=3

n=8

n=2

n=40

Correct answer:

n=2

Explanation:

To find the value of when,

$6\times 7\equiv n\ (\text{mod } 5)$ first multiply six and seven together.

$6\times 7=42$

Now, recall that mod means the remainder after division occurs.

In this case

$6\times 7\equiv n\ (\text{mod } 5)$

$5)\overline{42\ }$

--------------

Therefore, the remainder is two.

n=2

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Example Question #5 : Finite Mathematics

The following tables describe matrix operations.

$\begin{tabular}{ c|cc } \$ & * & \% \\ \hline * & \%& * \\ \% & * & * \\ \end{tabular}$ $\begin{tabular}{ c|cc } \# & \%& * \\ \hline \% & *& \% \\ * & \% & * \\ \end{tabular}$

Calculate the following using the tables from above.

$(\%\ \#\ \%)\ \$\ \%$

Possible Answers:

*^2

$\%$

$\#$

$\$$

Correct answer:

Explanation:

This question is testing the matrix operation. Remember to use order of operations and perform the algebraic operation that is inside the parentheses first.

$(\%\ \#\ \%)\ \$\ \%$

First look at the # table.

$\begin{tabular}{ c|cc } \# & \%& * \\ \hline \% & *& \% \\ * & \% & * \\ \end{tabular}$

Multiplying % by % results in *.

Now go to the $ table and multiply * by %.

$\begin{tabular}{ c|cc } \$ & * & \% \\ \hline * & \%& * \\ \% & * & * \\ \end{tabular}$

$*\ \$\ \%=*$

Therefore,

$(\%\ \#\ \%)\ \$\ \%=*$

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Example Question #6 : Finite Mathematics

Find the value of when,

$3\times 9\equiv n\ (\text{mod } 4)$ .

Possible Answers:

n=0

n=4

n=2

n=6

n=3

Correct answer:

n=3

Explanation:

To find the value of when,

$3\times 9\equiv n\ (\text{mod } 4)$ first multiply three and nine together.

$3\times 9=27$

Now, recall that mod means the remainder after division occurs.

In this case

$3\times 9\equiv n\ (\text{mod } 4)$

$4)\overline{27\ }$

--------------

Therefore, the remainder is three.

n=3

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Example Question #1 : Finite Mathematics

$A = \begin{bmatrix} 3 & -1 \\ -2 & 4 \end{bmatrix}$

Give the determinant of .

Possible Answers:

Correct answer:

Explanation:

The determinant of a two-by-two matrix

$A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

can be found by evaluating the expression

$\det A = a_{11}a_{22} - a_{12}a_{23}$

Substitute the corresponding elements to get

$\det A = 3(4)- (-1)(-2) = 12 - 2 = 10$ .

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Example Question #1 : Systems Of Linear Equations: Matrices

Consider the system of linear equations:

x + 8y = 120

-4x + 32y = 240

What kind of system is this?

Possible Answers:

Consistent and independent

Dependent

Inconsistent

Correct answer:

Consistent and independent

Explanation:

One way to identify whether the matrix is consistent and independent is to form a matrix of its variable coefficients, and calculate its determinant. The matrix is

$A = \begin{bmatrix} 1 & 8 \\ -4 & 32 \end{bmatrix}$

The determinant of a two-by-two matrix

$A= \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$

can be found by evaluating the expression

$\det A = a_{11}a_{22} - a_{12}a_{23}$

Substitute the corresponding elements to get

$\det A = 1(32)- 8(-4) = 32 -(-32) = 64$

Since $\det A \ne 0$ ,

it follows that the system is consistent and independent.

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Example Question #1 : Systems Of Linear Equations: Matrices

$B = \begin{bmatrix} 3 & 2 \\ 4 & 3 \end{bmatrix}$

Which of the following is equal to $B^{-1}$ ?

Possible Answers:

$\begin{bmatrix} -3 & 4 \\ 2 &- 3 \end{bmatrix}$

$\begin{bmatrix} 3 & 4 \\ 2 & 3 \end{bmatrix}$

$\begin{bmatrix} -3& 2 \\ 4 & -3 \end{bmatrix}$

$\begin{bmatrix} -3& -2 \\ -4 & -3 \end{bmatrix}$

$\begin{bmatrix} 3& -2 \\ -4 & 3 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 3& -2 \\ -4 & 3 \end{bmatrix}$

Explanation:

$B^{-1}$ , the inverse of a two-by-two matrix

$B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}$ ,

can be calculated as follows:

$B^{-1} = \frac{1}{D} \begin{bmatrix} b_{22} & -b_{12} \\ -b_{21} & b_{11} \end{bmatrix}$ ,

where $D = b_{11}b_{22} - b_{12} b_{21}$ .

Setting each of the values accordingly,

D = 3(3)- 2(4) = 9 - 8 = 1

$B^{-1} = \frac{1}{1} \begin{bmatrix} 3& -2 \\ -4 & 3 \end{bmatrix}$ , or

$B^{-1} = \begin{bmatrix} 3& -2 \\ -4 & 3 \end{bmatrix}$ .

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Example Question #11 : Finite Mathematics

$A = \begin{bmatrix} 2 & 1 & 3 \\ - 1 & -2 & -3 \end{bmatrix}$

refers to the two-by-two identity matrix.

Which of the following expressions is equal to A+ I ?

Possible Answers:

$\begin{bmatrix} 3& 1 & 3 \\ - 1 & -1 & -3 \end{bmatrix}$

$\begin{bmatrix} 2 & 2 & 3 \\ - 1 & -2 & -2 \end{bmatrix}$

$\begin{bmatrix} 2 & 1 \\ - 1 & -2 \end{bmatrix}$

A+ I is undefined.

$\begin{bmatrix} 3 & 1 & 3 \\ - 1 & -2 & -2 \end{bmatrix}$

Correct answer:

A+ I is undefined.

Explanation:

For the sum of two matrices to be defined, they must have the same number of rows and columns. is a matrix with three columns; since, in this problem, refers to the two-by-two identity matrix

$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ ,

has two columns. Since the number of columns differs, A+ I is undefined.

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Example Question #1 : Systems Of Linear Equations: Matrices

is a three-by-four matrix.

Which must be true?

Possible Answers:

has four rows.

has three rows.

None of the statements in the other choices must be true.

Correct answer:

has three rows.

Explanation:

The product of two matrices and , where has rows and columns and has rows and columns, is a matrix with rows and columns. It follows that must have the same number of rows as . Since has three rows, so does . Nothing can be inferred about the number of rows of .

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Example Question #1 : Systems Of Linear Equations: Matrices

$K = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$

True or false:

$7K = \begin{bmatrix} 7 & -14 & 21 \\ -4 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$

Possible Answers:

False

True

Correct answer:

False

Explanation:

The product of any scalar value and a matrix is the matrix formed when each entry in the matrix is multiplied by that scalar. Thus, if

$K = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$

then

$7K = \begin{bmatrix} 7 (1) & 7 ( -2 )& 7(3) \\ 7(-4) & 7(2) & 7(1) \\ 7(2) & 7(1) & 7(3) \end{bmatrix}$

$= \begin{bmatrix} 7& -14 & 21 \\ -28 & 14 & 7 \\ 14 & 7 & 21 \end{bmatrix}$

This is not equal to the matrix

$\begin{bmatrix} 7 & -14 & 21 \\ -4 & 2 & 1 \\ 2 & 1 & 3 \end{bmatrix}$ ,

since the entries in the second and third rows differ. The statement is false.

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Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

$\begin{bmatrix} 14 \\ 25 \end{bmatrix} = \begin{bmatrix} x + y \\ x - y \end{bmatrix}$

True or false: there is no solution (x,y) that makes this matrix equation true.

Possible Answers:

True

False

Correct answer:

False

Explanation:

For two matrices to be equal, two conditions must hold:

1) The matrices must have the same dimension. This can be seen to be the case, since both matrices have two rows and one column.

2) All corresponding entries must be equal. For this to happen, it must hold that

x+y=14

x-y=25

This is a system of two equations in two variables, which can be solved as follows:

Add both sides of the equations:

$\begin{matrix} x+y=14 \\ \underline{x-y=25} \\ 2x\; \; \; \; \; = 39 \end{matrix}$

It follows that

x = 19.5

Substitute back:

x+y=14

19.5+y=14

y = -5.5

Thus, there exists a solution to the system, and, consequently, to the original matrix equation. The statement is therefore false.

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Finite Mathematics : Systems of Linear Equations: Matrices

Study concepts, example questions & explanations for Finite Mathematics

All Finite Mathematics Resources

Example Questions

Example Question #4 : Finite Mathematics

Example Question #5 : Finite Mathematics

Example Question #6 : Finite Mathematics

Example Question #1 : Finite Mathematics

Example Question #1 : Systems Of Linear Equations: Matrices

Example Question #1 : Systems Of Linear Equations: Matrices

Example Question #11 : Finite Mathematics

Example Question #1 : Systems Of Linear Equations: Matrices

Example Question #1 : Systems Of Linear Equations: Matrices

Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

All Finite Mathematics Resources

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