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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

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6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #115 : Matrices

Evaluate the following matrix operation:

$2\mathbf{A}+4\mathbf{B}$

where

$\mathbf{A}=\begin{pmatrix} 1&4 \\ -3&0 \end{pmatrix}, \, \;\mathbf{B}=\begin{pmatrix} 2&-2 \\ 1&-1 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 8&12 \\ -10&-2 \end{pmatrix}$

$\begin{pmatrix} 10&0 \\ -2&-4 \end{pmatrix}$

$\begin{pmatrix} -6&16 \\ -10&4 \end{pmatrix}$

$\begin{pmatrix} -1&6 \\ -4&1 \end{pmatrix}$

$\begin{pmatrix} 2&-8 \\ -2&-1 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} 10&0 \\ -2&-4 \end{pmatrix}$

Explanation:

$2\mathbf{A}+4\mathbf{B}=2\begin{pmatrix} 1&4 \\ -3&0 \end{pmatrix}+4 \begin{pmatrix} 2&-2 \\ 1&-1 \end{pmatrix}= \begin{pmatrix} 2&8 \\ -6&0 \end{pmatrix}+ \begin{pmatrix} 8&-8 \\ 4&-4 \end{pmatrix}$

$=\begin{pmatrix} 2+8&8-8 \\ -6+4&0-4 \end{pmatrix}= \begin{pmatrix} 10&0 \\ -2&-4 \end{pmatrix}$

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Example Question #2711 : Calculus 3

Find the determinant of the matrix A:

$\mathbf{A}= \begin{pmatrix} 7&4 \\ 5&2 \end{pmatrix}$

Possible Answers:

-6

-24

Correct answer:

-6

Explanation:

The determinant of a matrix

$\begin{pmatrix} a&b \\ c&d \end{pmatrix}$

is defined as:

ad-bc

Here, that becomes:

(7)(2)-4(5)=14-20=-6

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Example Question #711 : Vectors And Vector Operations

$A=\begin{pmatrix} 1&4 \\ 5&7 \end{pmatrix}$

What is A^t ?

Possible Answers:

$A^t=\begin{pmatrix} 5&1 \\ 7& 4 \end{pmatrix}$

$A^t=\begin{pmatrix} 4&7 \\ 1& 5 \end{pmatrix}$

$A^t=\begin{pmatrix} 1&5 \\ 4& 7 \end{pmatrix}$

$A^t=\begin{pmatrix} 1&-4 \\ -5& 7 \end{pmatrix}$

Correct answer:

$A^t=\begin{pmatrix} 1&5 \\ 4& 7 \end{pmatrix}$

Explanation:

To find A^t , we write the rows of as columns.

Hence,

$A^t=\begin{pmatrix} 1&5 \\ 4& 7 \end{pmatrix}$

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Example Question #712 : Vectors And Vector Operations

$A=\begin{pmatrix} 4&3 \\ 2&5 \end{pmatrix}$

What is A^t ?

Possible Answers:

$A^t=\begin{pmatrix} 2&4 \\ 5&3 \end{pmatrix}$

$A^t=\begin{pmatrix} 4&2 \\ 3& 5 \end{pmatrix}$

$A^t=\begin{pmatrix} 4&-3 \\ -2&5 \end{pmatrix}$

$A^t=\begin{pmatrix} 3&5 \\ 4&2 \end{pmatrix}$

Correct answer:

$A^t=\begin{pmatrix} 4&2 \\ 3& 5 \end{pmatrix}$

Explanation:

To find A^t , we write the rows of as columns.

Hence,

$A^t=\begin{pmatrix} 4&2 \\ 3& 5 \end{pmatrix}$

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Example Question #2712 : Calculus 3

Find the determinant of the matrix.

$\begin{bmatrix} 1& 4&5 \\ 2& 3&4 \\ 0&1 &1 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

The formula for the determinant of a 3x3 matrix

$A=\begin{bmatrix} a&b &c \\ d &e & f\\ g&h &i \end{bmatrix}$ is

$\left | A\right |=a(ei-hf)-b(di-gf)+c(dh-ge)$ .

Using the matrix we were given, we get

1(3-4)-4(2-0)+5(2-0)=1 .

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Example Question #714 : Vectors And Vector Operations

Perform the matrix operation.

$\begin{bmatrix} 2& 3\\6 & 7 \end{bmatrix}+\begin{bmatrix} 1& 5\\ 0& 1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 9&1 \\ 2&3 \end{bmatrix}$

$\begin{bmatrix} 3& 8\\ 6&8 \end{bmatrix}$

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

$\begin{bmatrix} 5&6 \\ 1&2 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 3& 8\\ 6&8 \end{bmatrix}$

Explanation:

The formula for adding a pair of 2x2 matrices is

$\begin{bmatrix} a& b\\ c&d \end{bmatrix}+\begin{bmatrix} e&f \\ g &h \end{bmatrix}=\begin{bmatrix} (a+e)&(b+f) \\ (c+g)& (d+h) \end{bmatrix}$ .

Using the matrices in the problem statement, we get

$\begin{bmatrix} (2+1)&(3+5) \\ (6+0)&(7+1) \end{bmatrix}=\begin{bmatrix} 3&8 \\ 6& 8 \end{bmatrix}$

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Example Question #715 : Vectors And Vector Operations

Perform the following matrix operation:

$\begin{bmatrix}1 & 5 \\ 2 & 4 \end{bmatrix}-\begin{bmatrix}2 & 4\\ 9 & 1 \end{bmatrix}$

Possible Answers:

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

$\begin{bmatrix}-1 &1 \\ -7 & 3 \end{bmatrix}$

$\begin{bmatrix}-3 &0 \\ -7 & 3 \end{bmatrix}$

$\begin{bmatrix}-2 &1 \\ -5 & 0 \end{bmatrix}$

Correct answer:

$\begin{bmatrix}-1 &1 \\ -7 & 3 \end{bmatrix}$

Explanation:

To perform the matrix operation, you must follow the formula

$\begin{bmatrix}a &b \\c & d \end{bmatrix}-\begin{bmatrix} e&f \\g & h \end{bmatrix}=\begin{bmatrix}{a-e} & b-f\\c-g & d-h \end{bmatrix}$ .

Using the values we have, this becomes

$\begin{bmatrix}1-2 & 5-4\\ 2-9&4-1 \end{bmatrix}$

which simplified is

$\begin{bmatrix}-1 &1 \\ -7 & 3 \end{bmatrix}$

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Example Question #716 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix:

$\begin{bmatrix}1 &2 &4 \\5 & 0&3 \\ 4&1 &0 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

The formula for the determinant is as follows:

$\left | A\right |=a(ei-fh)-b(di-fg)+c(dh-eg)$ ,

where

$A=\begin{bmatrix}a &b &c \\ d& e& f\\ g &h &i \end{bmatrix}$ .

Using the matrix in the problem, we get

1(0-3)-2(0-12)+4(5-0)=-3+24+20=41

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Example Question #2713 : Calculus 3

$A=\begin{pmatrix} 1\\ 2\\ 3 \end{pmatrix}$

What is A^t ?

Possible Answers:

$A^t=\begin{pmatrix} 1&2 &3 \end{pmatrix}$

$A=\begin{pmatrix} 3\\1\\ 2 \end{pmatrix}$

$A=\begin{pmatrix} 2\\3\\ 1 \end{pmatrix}$

Correct answer:

$A^t=\begin{pmatrix} 1&2 &3 \end{pmatrix}$

Explanation:

To find

A^t , we write the rows of

as columns.

Hence,

$A^t=\begin{pmatrix} 1&2 &3 \end{pmatrix}$

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Example Question #2714 : Calculus 3

Find the determinant of the 3x3 matrix $\begin{bmatrix} 3&1 &4 \\ 4&9 &-4 \\ 5&8 &0 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

The formula for the determinant of a 3x3 matrix $A=\begin{bmatrix} a&b &c \\ d& e&f \\ g&h &i \end{bmatrix}$ is $\left | A\right |=a(ei-hf)-b(di-gf)+c(dh-ge)$ . Using the matrix in the problem statement, we get 3(0+32)-1(0+20)+4(32-45)=96-20-62=14

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #115 : Matrices

Example Question #2711 : Calculus 3

Example Question #711 : Vectors And Vector Operations

Example Question #712 : Vectors And Vector Operations

Example Question #2712 : Calculus 3

Example Question #714 : Vectors And Vector Operations

Example Question #715 : Vectors And Vector Operations

Example Question #716 : Vectors And Vector Operations

Example Question #2713 : Calculus 3

Example Question #2714 : Calculus 3

All Calculus 3 Resources

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