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Calculus 3 : Matrices

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Example Questions

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 #142 : Matrices

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 #143 : Matrices

Find the determinant of the matrix:

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

Possible Answers:

4y^2-x^3y

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

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 #144 : Matrices

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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Example Question #145 : Matrices

Find the determinant of the 2x2 matrix $\begin{bmatrix} 4&2 \\ 8&3 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

To find the determinant of a 2x2 matrix $a=\begin{bmatrix} b& c\\ x&y \end{bmatrix}$ ,

we use the formula

determinant=by-xc

Using the matrix from the problem statement, we get

(4*3)-(8*2)=-4

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Example Question #146 : Matrices

Find the determinant of the 2x2 matrix $\begin{bmatrix} 5&-1\\ 12&3\end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

To find the determinant of a 2x2 matrix $a=\begin{bmatrix} b& c\\ x&y \end{bmatrix}$ ,

we use the formula

determinant=by-xc

Using the matrix from the problem statement, we get

(5*3)-(12*-1)=27

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Example Question #147 : Matrices

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

Possible Answers:

Correct answer:

Explanation:

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

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

Using the matrix from the problem statement and applying, we get

4(6-14)-3(-18-0)+5(-6-0)=-8

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Example Question #148 : Matrices

Find the determinant of the 3x3 matrix $\begin{bmatrix} -6& 0&1 \\ -2&4 &9\\ 2&5&0\end{bmatrix}$

Possible Answers:

300

222

135

Correct answer:

222

Explanation:

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

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

Using the matrix from the problem statement and applying, we get

-6(0-40)-0(0-18)+1(-10-8)=222

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Example Question #149 : Matrices

Find the determinant of the 2x2 matrix $\begin{bmatrix} 4&5x \\ 2z&12 \end{bmatrix}$

Possible Answers:

14-10xz

48-10xz

40-10xz

48+10xz

Correct answer:

48-10xz

Explanation:

To find the determinant of a 2x2 matrix $\begin{bmatrix} a&b \\ c& d \end{bmatrix}$ we use the definition:
determinant=ad-cb

Applying to the matrix from the problem statement, we get

determinant=(4*12)-(2z*5x)=48-10xz

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Example Question #150 : Matrices

Find the determinant of the 2x2 matrix $\begin{bmatrix} 3& 6\\ 1&7 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

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

determinant=xq-zy

Using the matrix from the problem statement, we get

(7*3)-(6*1)=15

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Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #141 : Matrices

Example Question #142 : Matrices

Example Question #143 : Matrices

Example Question #144 : Matrices

Example Question #145 : Matrices

Example Question #146 : Matrices

Example Question #147 : Matrices

Example Question #148 : Matrices

Example Question #149 : Matrices

Example Question #150 : Matrices

All Calculus 3 Resources

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