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Calculus 3 : Vectors and Vector Operations

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Example Questions

Example Question #701 : Vectors And Vector Operations

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 4&-9 \\-2 &-9 \end{vmatrix}$ and $b=\begin{vmatrix} -6\\-3 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 13\\39 \end{vmatrix}$

$\begin{vmatrix} -6\\-48 \end{vmatrix}$

$\begin{vmatrix} 4\\-12 \end{vmatrix}$

$\begin{vmatrix} 3\\39 \end{vmatrix}$

$\begin{vmatrix} -3\\-48 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} 3\\39 \end{vmatrix}$

Explanation:

In order to multiply two matrices, $A\times b$ , the respective dimensions of each must be of the form $m \times n$ and $n \times p$ to create an $m\times p$ (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

$(A\times b \neq b \times A)$

For a multiplication of the form

$\begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}$

The resulting matrix is

$\begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}$

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix} 4&-9 \\-2 &-9 \end{vmatrix}$ and $b=\begin{vmatrix} -6\\-3 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -24+27\\ 12+27\end{vmatrix}=\begin{vmatrix} 3\\39 \end{vmatrix}$

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Example Question #701 : Vectors And Vector Operations

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 6&-9 \\6 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 2\\ 3\end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -15\\9 \end{vmatrix}$

$\begin{vmatrix} 20\\6 \end{vmatrix}$

$\begin{vmatrix} -3\\19 \end{vmatrix}$

$\begin{vmatrix} 14\\-5 \end{vmatrix}$

$\begin{vmatrix} 4\\5 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -15\\9 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix} 6&-9 \\6 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 2\\ 3\end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 12-27\\ 12-3\end{vmatrix}=\begin{vmatrix} -15\\9 \end{vmatrix}$

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Example Question #109 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} -9&-1 \\4 &-5 \end{vmatrix}$ and $b=\begin{vmatrix}-9\\-5 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 36\\ 75\end{vmatrix}$

$\begin{vmatrix} 86\\ -11\end{vmatrix}$

$\begin{vmatrix} 43\\ -7\end{vmatrix}$

$\begin{vmatrix} 22\\ 19\end{vmatrix}$

$\begin{vmatrix} 19\\ 73\end{vmatrix}$

Correct answer:

$\begin{vmatrix} 86\\ -11\end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix} -9&-1 \\4 &-5 \end{vmatrix}$ and $b=\begin{vmatrix}-9\\-5 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix}81+5 \\-36+25 \end{vmatrix}=\begin{vmatrix} 86\\ -11\end{vmatrix}$

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Example Question #110 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 0&-8 \\-7 &8 \end{vmatrix}$ and $b=\begin{vmatrix} 5\\5 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -8\\5 \end{vmatrix}$

$\begin{vmatrix} -8\\-25 \end{vmatrix}$

$\begin{vmatrix} -40\\5 \end{vmatrix}$

$\begin{vmatrix} 16\\25 \end{vmatrix}$

$\begin{vmatrix} 16\\-5 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -40\\5 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix} 0&-8 \\-7 &8 \end{vmatrix}$ and $b=\begin{vmatrix} 5\\5 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 0-40\\-35+40 \end{vmatrix}=\begin{vmatrix} -40\\5 \end{vmatrix}$

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Example Question #2701 : Calculus 3

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}-8 &1 &0 \\-8 &-5 &-9 \\2 &-7 &7 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\-5 \\-8 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -39\\-115 \\82 \end{vmatrix}$

$\begin{vmatrix} 86\\25 \\44 \end{vmatrix}$

$\begin{vmatrix} 11\\-5 \\4 \end{vmatrix}$

$\begin{vmatrix} -23\\18 \\-42 \end{vmatrix}$

$\begin{vmatrix} -37\\65 \\-13 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -37\\65 \\-13 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix}-8 &1 &0 \\-8 &-5 &-9 \\2 &-7 &7 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\-5 \\-8 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -32-5+0\\-32+25+72 \\8+35-56 \end{vmatrix}=\begin{vmatrix} -37\\65 \\-13 \end{vmatrix}$

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Example Question #2702 : Calculus 3

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}9 &7 &9 \\7 &-5 &-6 \\-1 &3 &-9 \end{vmatrix}$ and $b=\begin{vmatrix} -7\\-7 \\8 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix}40 \\62 \\-86 \end{vmatrix}$

$\begin{vmatrix}-40 \\-62 \\86 \end{vmatrix}$

$\begin{vmatrix}-40 \\62 \\-86 \end{vmatrix}$

$\begin{vmatrix}-40 \\-62 \\-86 \end{vmatrix}$

$\begin{vmatrix}40 \\62 \\86 \end{vmatrix}$

Correct answer:

$\begin{vmatrix}-40 \\-62 \\-86 \end{vmatrix}$

Explanation:

$(A\times b \neq b \times A)$

For a multiplication of the form

The resulting matrix is

The notation may be daunting but numerical examples may elucidate.

We're told that

$A=\begin{vmatrix}9 &7 &9 \\7 &-5 &-6 \\-1 &3 &-9 \end{vmatrix}$ and $b=\begin{vmatrix} -7\\-7 \\8 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -63-49+72\\-49+35-48 \\7-21-72 \end{vmatrix}=\begin{vmatrix}-40 \\-62 \\-86 \end{vmatrix}$

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Example Question #2701 : Calculus 3

Calculate the determinant of .

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

Possible Answers:

$\det(A)=-2$

$\det(A)=1$

$\det(A)=2$

$\det(A)=-1$

$\det(A)=0$

Correct answer:

$\det(A)=-2$

Explanation:

In order to find the determinant, we need to multiply the main diagonal components and then subtract the off main diagonal components.

$\det(A)=1(4)-(3)(2)=4-6=-2$

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Example Question #704 : Vectors And Vector Operations

Find the product of the two matrices:

$\mathbf{A}\mathbf{B}$

Where

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

and

$\mathbf{B}=\begin{pmatrix} -1&4 \\ 0&5 \end{pmatrix}$

Possible Answers:

$\begin{pmatrix} 5&10 \\ -9&-10 \end{pmatrix}$

$\begin{pmatrix} 3&-2 \\ 17&-2 \end{pmatrix}$

$\begin{pmatrix} -3&17 \\ -2&-2 \end{pmatrix}$

$\begin{pmatrix} 5&-9 \\ 10&-10 \end{pmatrix}$

Correct answer:

$\begin{pmatrix} -3&17 \\ -2&-2 \end{pmatrix}$

Explanation:

$\mathbf{AB}=\begin{pmatrix} 3&1 \\ 2&-2 \end{pmatrix}\textup{}$ $\begin{pmatrix} -1&4 \\ 0&5 \end{pmatrix}$

$=\begin{pmatrix} (3)(-1)+(1)(0)&(3)(4)+(1)(5) \\ (2)(-1)+(-2)(0)&(2)(4)+(-2)(5) \end{pmatrix}$

$=\begin{pmatrix} (-3+0)&(12+5) \\ (-2+0)&(8-10) \end{pmatrix}=\begin{pmatrix} -3&17 \\ -2&-2 \end{pmatrix}$

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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 #111 : Matrices

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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Calculus 3 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #701 : Vectors And Vector Operations

Example Question #701 : Vectors And Vector Operations

Example Question #109 : Matrices

Example Question #110 : Matrices

Example Question #2701 : Calculus 3

Example Question #2702 : Calculus 3

Example Question #2701 : Calculus 3

Example Question #704 : Vectors And Vector Operations

Example Question #115 : Matrices

Example Question #111 : Matrices

All Calculus 3 Resources

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