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Calculus 3 : Matrices

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Example Questions

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 #121 : Matrices

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

What is A^t ?

Possible Answers:

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

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

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

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 #122 : Matrices

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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Example Question #719 : Vectors And Vector Operations

Evaluate the determinant of the matrix.

$A=\begin{pmatrix} 1&3 \\ 8&6 \end{pmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of the 2x2 matrix

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

is defined as

ad-bc

For the matrix in this problem the determinant is

=1(6)-8(3)

=6-24

=-18

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Example Question #123 : Matrices

Find the determinant of the matrix:

$\begin{vmatrix} 3&x \\ \cos(y)&z \end{vmatrix}$

Possible Answers:

$3z-x\cos(y)$

$3x-z\cos(y)$

$3z-\cos(y)$

$3x+x\cos(y)$

Correct answer:

$3z-x\cos(y)$

Explanation:

To find the determinant of the matrix, we multiply across, from top left to bottom right, and then subtract the product of multiplying across from top right to bottom left:

$\begin{vmatrix} a& b\\ c& d \end{vmatrix} = ad-bc$

Our answer is

$3z-x\cos(y)$

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Example Question #124 : Matrices

Find the determinant of the 2x2 matrix $\begin{bmatrix} 4& 5\\ 1& 20 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

The formula for the determinant of a 2x2 matrix $a=\begin{bmatrix} a&b \\ c& d \end{bmatrix}$ is $\left | a\right |=ad-cb$ . Using the matrix from the problem statement, we get (20*4)-(1*5)=75

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Example Question #125 : Matrices

Find the determinant of the 2x2 matrix $\begin{bmatrix} 1&14 \\ 2& 5 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

The formula foe the determinant of a 2x2 matrix $\begin{bmatrix} a&b \\ c& d \end{bmatrix}$ is ad-cb . Using the matrix from the problem statement, we get (1*5)-(2*14)=5-28=-23

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Example Question #126 : Matrices

Find the determinant of the given matrix:

$\begin{vmatrix} x& y\\ \sin(z)&\cos(x) \end{vmatrix}$

Possible Answers:

$x\cos(x)+y\sin(z)$

$x\cos(x)-y\sin(z)$

$\left \langle x\cos(x), -y\sin(z) \right \rangle$

Correct answer:

$x\cos(x)-y\sin(z)$

Explanation:

The determinant of the matrix is found by multiplying from top left to bottom right, and then subtracting the product from top right to bottom left (for example, the determinant of $\begin{vmatrix} a&b \\ c& d \end{vmatrix}=ad-bc$ )

Our answer is

$x\cos(x)-y\sin(z)$

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Example Question #127 : Matrices

Find the determinant of the 2x2 matrix $\begin{bmatrix} 4& x\\ 2y& z \end{bmatrix}$

Possible Answers:

4z-2yx

xy-4xyz

4-2xy

2x-3y

Correct answer:

4z-2yx

Explanation:

To find the determinant of a 2x2 matrix $a=\begin{bmatrix} a&b \\ c&d \end{bmatrix}$ we use the formula for the determinant, which is (a*d)-(c*b) . Plugging in the values from the problem statement, we find that the determinant of the matrix is (4*z)-(2y*x)=4x-2yx

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Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #715 : Vectors And Vector Operations

Example Question #716 : Vectors And Vector Operations

Example Question #121 : Matrices

Example Question #122 : Matrices

Example Question #719 : Vectors And Vector Operations

Example Question #123 : Matrices

Example Question #124 : Matrices

Example Question #125 : Matrices

Example Question #126 : Matrices

Example Question #127 : Matrices

All Calculus 3 Resources

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