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Calculus 3 : Calculus 3

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Example Questions

Example Question #729 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix $\begin{bmatrix} i&j &k \\ 3&-4 &1 \\ 2&0 &5 \end{bmatrix}$

Possible Answers:

$\left \langle 20,15,8\right \rangle$

$\left \langle -20,-13,8\right \rangle$

$\left \langle -20,13,8\right \rangle$

$\left \langle 4,-10,8\right \rangle$

Correct answer:

$\left \langle -20,-13,8\right \rangle$

Explanation:

To find the determinant of the 3x3 matrix $\begin{bmatrix} i& j& k\\ a&b &c \\ x&y &z \end{bmatrix}$ , we use the formula:

determinant=i(bz-yc)-j(az-xc)+k(ay-xb) . Using the matrix from the problem statement, we get:

$i(-20-0)-j(15-2)+k(0+8)=\left \langle -20,-13,8\right \rangle$

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Example Question #730 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix $\begin{bmatrix} i&j &k \\ 2&1 &1 \\ 6&6 &7 \end{bmatrix}$

Possible Answers:

$\left \langle 2,1,8\right \rangle$

$\left \langle 1,-8,6\right \rangle$

$\left \langle 2,-8,9\right \rangle$

$\left \langle 1,8,6\right \rangle$

Correct answer:

$\left \langle 1,-8,6\right \rangle$

Explanation:

To find the determinant of the 3x3 matrix $\begin{bmatrix} i& j& k\\ a&b &c \\ x&y &z \end{bmatrix}$ , we use the formula:

determinant=i(bz-yc)-j(az-xc)+k(ay-xb) . Using the matrix from the problem statement, we get:

$i(7-6)-j(14-6)+k(12-6)=\left \langle 1,-8,6\right \rangle$

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Example Question #137 : Matrices

Perform the matrix operation $\begin{bmatrix} 4&2 &0 &1 \\ 3&2 &1 &7 \\ 3&4 &5 &1 \\ 2&3 &0 & 5 \end{bmatrix}+\begin{bmatrix} 4&2 &8 &0 \\ 5&3 &1 &6 \\ 5&11 &10 &2 \\ 4&6 &9 &1 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 1&4 &0 &0 \\ 15&3 &1 &42 \\ 15&44 &50 &2 \\ 7&18 &0 &1 \end{bmatrix}$

$\begin{bmatrix} 16&4 &0 &1 \\ 15&6 &1 &4 \\ 5&19 &50 &2 \\ 8&18 &0 &5 \end{bmatrix}$

$\begin{bmatrix} 1&4 &0 &0 \\ 15&6 &1 &4 \\ 15&44 &5 &2 \\ 8&8 &0 &5 \end{bmatrix}$

$\begin{bmatrix} 8&4 &8 &1 \\ 8&5 &2 &13 \\ 8&15 &15 &3 \\ 6&9 &9 &6 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 8&4 &8 &1 \\ 8&5 &2 &13 \\ 8&15 &15 &3 \\ 6&9 &9 &6 \end{bmatrix}$

Explanation:

To do matrix summation, you apply the following formula, where a and b are 4x4 matrices: $\begin{bmatrix} a&b &c &d \\ e&f &g &h \\ i&j &k &l \\ m&n & o&p \end{bmatrix}+\begin{bmatrix} q& r& s&t \\ u&v &w &x \\ y&z &a_1 &b_1 \\ c_1&d_1 &e_1 &f_1 \end{bmatrix}=\begin{bmatrix} a+q&b+r &c+s &d+t \\ e+u&f+v &g+w &h+x \\ i+y&j+z &k+a_{1} &l+b_{1} \\ m+c_{1}&n+d_{1} &o+e_{1} &p+f_{1} \end{bmatrix}$

Using this formula, we get $\begin{bmatrix} 8&4 &8 &1 \\ 8&5 &2 &13 \\ 8&15 &15 &3 \\ 6&9 &9 &6 \end{bmatrix}$

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Example Question #731 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix given: $\begin{bmatrix} 1&4 &7 \\ 3&7 &8 \\ 5&0 &6 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

To find the the determinant of a 3x3 matrix $a=\begin{bmatrix} a& b&c \\ d&e &f \\ h&i &j \end{bmatrix}$ , you apply the following formula

determinant=a(ej-if)-b(dj-hf)+c(di-he)

Applying to the matrix from the problem statement, we get

1(42-0)-4(18-40)+7(0-35)=-115

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Example Question #732 : Vectors And Vector Operations

Find the determinant of the 3x3 matrix given: $\begin{bmatrix} 2&0 &4 \\ 3&1 &6 \\ 5&2 &7 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

To find the the determinant of a 3x3 matrix $a=\begin{bmatrix} a& b&c \\ d&e &f \\ h&i &j \end{bmatrix}$ , you apply the following formula

determinant=a(ej-if)-b(dj-hf)+c(di-he)

Applying to the matrix from the problem statement, we get

2(7-12)-0(21-30)+4(6-5)=-6

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Example Question #2731 : Calculus 3

Find the determinant of the 2x2 matrix given: $\begin{bmatrix} 3&x \\ 7&2 \end{bmatrix}$

Possible Answers:

6-7x

2x+3

3x+9

6+7x

Correct answer:

6-7x

Explanation:

To find the determinant of a 2x2 matrix $a=\begin{bmatrix} x&y \\ z&q \end{bmatrix}$ , we use the formula:

detemrinant=xq-zy

Using the vector from the problem statement, we get:

(3*2)-(7*x)=6-7x

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Example Question #141 : Matrices

Evaluate the matrix operation $\begin{bmatrix} 4&7 \\ 1&-3 \end{bmatrix}+\begin{bmatrix} 9&-4 \\ -2& -5 \end{bmatrix}$

Possible Answers:

$\begin{bmatrix} 13&3 \\ -1&-8 \end{bmatrix}$

$\begin{bmatrix} 13&-3 \\ -1&-8 \end{bmatrix}$

$\begin{bmatrix} 15&3 \\ -1&-8 \end{bmatrix}$

$\begin{bmatrix} 13&3 \\ -1&-4 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 13&3 \\ -1&-8 \end{bmatrix}$

Explanation:

To find the sum of 2 2x2 matrices, we use the formula:

$\begin{bmatrix} a&b \\ c&d \end{bmatrix}+\begin{bmatrix} w&x \\ y& z \end{bmatrix}=\begin{bmatrix} a+w& b+x\\ c+y&d+z \end{bmatrix}$

Applying to the matrices from the problem statement, we get:

$\begin{bmatrix} 4+9&7-4 \\ 1-2&-3-5 \end{bmatrix}=\begin{bmatrix} 13&3 \\ -1&-8 \end{bmatrix}$

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Example Question #2732 : Calculus 3

Find the determinant of the 2x2 matrix given: $\begin{bmatrix} 4&y \\ z&8 \end{bmatrix}$

Possible Answers:

32-yz

32+yz

8y-4z

4y-8z

Correct answer:

32-yz

Explanation:

To find the determinant of a 2x2 matrix $a=\begin{bmatrix} x&y \\ z&q \end{bmatrix}$ , we use the formula:

detemrinant=xq-zy

Using the vector from the problem statement, we get:

(4*8)-(y*z)=32-yz

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Example Question #2733 : Calculus 3

Find the determinant of the matrix:

$\begin{vmatrix} 4&xy \\ x^2& y^2 \end{vmatrix}$

Possible Answers:

$\left \langle 4y^2, -x^3y \right \rangle$

4y^2+x^3y

4y^2-x^3y

Correct answer:

4y^2-x^3y

Explanation:

The determinant of a two-by-two matrix is given by

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

(The process is similar to the finding the cross product using a three-by-three matrix, where one moves from top left to bottom right, and then subtracts the result of moving from top right to bottom left).

Our final answer is 4y^2-x^3y .

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Example Question #2734 : Calculus 3

Find the determinant of the 3x3 matrix $\begin{bmatrix} 1&5 &9 \\ 2& 1&0 \\ 6&3 &2 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

To find the determinant of a 3x3 matrix $a=\begin{bmatrix} a&b &c \\ d& e&f \\ g& h&i \end{bmatrix}$ , we use the formula

determinant=a(ei-hf)-b(di-gf)+c(dh-ge)

Using the matrix from the problem statement, we get

determinant=1(2-0)-5(4-0)+9(6-6)=-18

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Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #729 : Vectors And Vector Operations

Example Question #730 : Vectors And Vector Operations

Example Question #137 : Matrices

Example Question #731 : Vectors And Vector Operations

Example Question #732 : Vectors And Vector Operations

Example Question #2731 : Calculus 3

Example Question #141 : Matrices

Example Question #2732 : Calculus 3

Example Question #2733 : Calculus 3

Example Question #2734 : Calculus 3

All Calculus 3 Resources

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