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Calculus 3 : Matrices

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Example Questions

Example Question #101 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}-4 & -4\\-3 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\ -1\end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -12\\-11 \end{vmatrix}$

$\begin{vmatrix} 12\\24 \end{vmatrix}$

$\begin{vmatrix} -8\\-6 \end{vmatrix}$

$\begin{vmatrix} -6\\-8 \end{vmatrix}$

$\begin{vmatrix} -11\\-12 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -12\\-11 \end{vmatrix}$

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

$A=\begin{vmatrix}-4 & -4\\-3 &-1 \end{vmatrix}$ and $b=\begin{vmatrix} 4\\ -1\end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -16+4\\-12+1 \end{vmatrix}=\begin{vmatrix} -12\\-11 \end{vmatrix}$

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Example Question #691 : Vectors And Vector Operations

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix}-6 & 0\\1 & -2\end{vmatrix}$ and $b=\begin{vmatrix} -1 \\ 0 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} 6\\0 \end{vmatrix}$

$\begin{vmatrix} 0\\-1 \end{vmatrix}$

$\begin{vmatrix} 0\\0 \end{vmatrix}$

$\begin{vmatrix} 6\\-1 \end{vmatrix}$

$\begin{vmatrix} 0\\-6 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} 6\\-1 \end{vmatrix}$

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

$A=\begin{vmatrix}-6 & 0\\1 & -2\end{vmatrix}$ and $b=\begin{vmatrix} -1 \\ 0 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 6+0\\-1+0 \end{vmatrix}=\begin{vmatrix} 6\\-1 \end{vmatrix}$

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Example Question #103 : Matrices

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

Possible Answers:

$\begin{vmatrix} 45\\-32 \end{vmatrix}$

$\begin{vmatrix} -66\\12 \end{vmatrix}$

$\begin{vmatrix} -44\\33 \end{vmatrix}$

$\begin{vmatrix} -66\\22 \end{vmatrix}$

$\begin{vmatrix} -56\\22 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -66\\22 \end{vmatrix}$

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

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

The resulting matrix product is then:

$A\times b = \begin{vmatrix}-10-56 \\ -18+40\end{vmatrix}=\begin{vmatrix} -66\\22 \end{vmatrix}$

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Example Question #104 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} 4& -9\\6 &9 \end{vmatrix}$ and $b=\begin{vmatrix} 6\\5 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -21\\81 \end{vmatrix}$

$\begin{vmatrix} -21\\-81 \end{vmatrix}$

$\begin{vmatrix} 21\\81 \end{vmatrix}$

$\begin{vmatrix} 21\\-81 \end{vmatrix}$

$\begin{vmatrix} 7\\-27 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -21\\81 \end{vmatrix}$

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

$A=\begin{vmatrix} 4& -9\\6 &9 \end{vmatrix}$ and $b=\begin{vmatrix} 6\\5 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} 24-45\\36+45 \end{vmatrix}=\begin{vmatrix} -21\\81 \end{vmatrix}$

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Example Question #105 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} -9&-3 \\2 &6 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\ -8\end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -2\\-16 \end{vmatrix}$

$\begin{vmatrix} -6\\-42 \end{vmatrix}$

$\begin{vmatrix} 12\\-13 \end{vmatrix}$

$\begin{vmatrix} 1\\26 \end{vmatrix}$

$\begin{vmatrix} -3\\-42 \end{vmatrix}$

Correct answer:

$\begin{vmatrix} -3\\-42 \end{vmatrix}$

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

$A=\begin{vmatrix} -9&-3 \\2 &6 \end{vmatrix}$ and $b=\begin{vmatrix} 3\\ -8\end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -27+24\\6-48 \end{vmatrix}=\begin{vmatrix} -3\\-42 \end{vmatrix}$

Report an Error

Example Question #106 : Matrices

Find the matrix product of $A\times b$ , where $A=\begin{vmatrix} -8& -9\\6 &3 \end{vmatrix}$ and $b=\begin{vmatrix} 7\\-4 \end{vmatrix}$

Possible Answers:

$\begin{vmatrix} -20\\30 \end{vmatrix}$

$\begin{vmatrix} 10\\30 \end{vmatrix}$

$\begin{vmatrix} 10\\-30 \end{vmatrix}$

$\begin{vmatrix} 10\\-20 \end{vmatrix}$

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

Correct answer:

$\begin{vmatrix} -20\\30 \end{vmatrix}$

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

$A=\begin{vmatrix} -8& -9\\6 &3 \end{vmatrix}$ and $b=\begin{vmatrix} 7\\-4 \end{vmatrix}$

The resulting matrix product is then:

$A\times b = \begin{vmatrix} -56+36\\42-12 \end{vmatrix}=\begin{vmatrix} -20\\30 \end{vmatrix}$

Report an Error

Example Question #107 : Matrices

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:

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

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

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

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

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

Correct answer:

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

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

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

The resulting matrix product is then:

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

Report an Error

Example Question #101 : Matrices

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:

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

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

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

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

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

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:

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

Report an Error

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:

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

Report an Error

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:

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

Report an Error

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Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #101 : Matrices

Example Question #691 : Vectors And Vector Operations

Example Question #103 : Matrices

Example Question #104 : Matrices

Example Question #105 : Matrices

Example Question #106 : Matrices

Example Question #107 : Matrices

Example Question #101 : Matrices

Example Question #109 : Matrices

Example Question #110 : Matrices

All Calculus 3 Resources

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