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Calculus 3 : Matrices

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Example Questions

Example Question #1 : Matrices

Calculate the determinant of Matrix .

$A=\begin{bmatrix} 1 & 2&3 \\ 0& 2& 1\\ 3& 4& 5 \end{bmatrix}$

Possible Answers:

$\det(A)=-2$

$\det(A)=-6$

$\det(A)=0$

$\det(A)=6$

$\det(A)=1$

Correct answer:

$\det(A)=-6$

Explanation:

In order to find the determinant of , we first need to copy down the first two columns into columns 4 and 5.

$A=\begin{bmatrix} 1 & 2&3 & 1 &2\\ 0& 2& 1&0 &2\\ 3& 4& 5 &3 &4\end{bmatrix}$

The next step is to multiply the down diagonals.

$A_d=1\cdot2\cdot5+2\cdot1\cdot 3+3\cdot0\cdot4=10+6+0=16$

The next step is to multiply the up diagonals.

$A_u=3\cdot2\cdot3+4\cdot1\cdot1+5\cdot0\cdot2=18+4=22$

The last step is to substract A_u from A_d .

$\det(A)=A_d-A_u=16-22=-6$

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Example Question #1 : Matrices

Calculate the determinant of Matrix .

$A=\begin{bmatrix} 0 & 1&5 \\ 10& -3& 2\\ 1& 2& -5 \end{bmatrix}$

Possible Answers:

$\det(A)=-38$

$\det(A)=0$

$\det(A)=38$

$\det(A)=67$

$\det(A)=167$

Correct answer:

$\det(A)=167$

Explanation:

In order to find the determinant of , we first need to copy down the first two columns into columns 4 and 5.

$A=\begin{bmatrix} 0 & 1&5 & 0 &1\\ 10& -3& 2&10 &-3\\ 1& 2& -5 &1 &2\end{bmatrix}$

The next step is to multiply the down diagonals.

$A_d=0\cdot-3\cdot-5+1\cdot2\cdot 1+5\cdot10\cdot2=0+2+100=102$

The next step is to multiply the up diagonals.

$A_u=1\cdot-3\cdot5+2\cdot2\cdot0+(-5)\cdot10\cdot1=-15+0-50=-65$

The last step is to substract A_u from A_d .

$\det(A)=A_d-A_u=102-(-65)=167$

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Example Question #2 : Matrices

Calculate the determinant of .

$A=\begin{bmatrix} 3 & 4\\ -2 & 10 \end{bmatrix}$

Possible Answers:

Correct answer:

Explanation:

All we need to do is multiply the main diagonal and substract it from the off diagonal.

$\det(A)=3\cdot10-4(-2)=30+8=38$

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Example Question #1 : Matrices

Which of the following is a way to represent the computation $<2,6,1> \cdot <5,2,4>$ using matrices?

Possible Answers:

$\begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5& 2 & 4 \end{bmatrix}$

$\begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5\\ 2\\ 4\\ \end{bmatrix}$

None of the other answers

All of the above answers

$\begin{bmatrix} 2 \\ 6 \\ 1 \end{bmatrix} \begin{bmatrix} 5 \\ 2 \\ 4 \end{bmatrix}$

Correct answer:

$\begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5\\ 2\\ 4\\ \end{bmatrix}$

Explanation:

This choice is the only pair with a well-defined operation; the others are meaningless in terms of matrix multiplication.

Computing this we get

$\begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5\\ 2\\ 4\\ \end{bmatrix} = (2)(5)+(6)(2)+(1)(4) =26$ .

Which is the same as

$<2,6,1> \cdot <5,2,4> = (2)(5)+(6)(2)+(1)(4) =26$ .

This operation is true for (real) -dimensional vectors in general;

$\begin{bmatrix} a & b & c \end{bmatrix} \begin{bmatrix} d\\ e\\ f\\ \end{bmatrix} = ad+be+cf$

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Example Question #2 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 1&2 \\ 5& 9\end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

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

For the matrix $A=\begin{vmatrix} 1&2 \\ 5& 9\end{vmatrix}$

The determinant is thus:

|A|=9-10=-1

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Example Question #1 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 4&2 \\-1 &3 \end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

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

For the matrix $A=\begin{vmatrix} 4&2 \\-1 &3 \end{vmatrix}$

The determinant is thus:

|A|=12-(-2)=14

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Example Question #7 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 5& 2\\ 3&1 \end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

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

For the matrix $A=\begin{vmatrix} 5& 2\\ 3&1 \end{vmatrix}$

The determinant is thus:

|A|=5-6=-1

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Example Question #8 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 4&4 \\4 &2 \end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

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

For the matrix $A=\begin{vmatrix} 4&4 \\4 &2 \end{vmatrix}$

The determinant is thus:

|A|=8-16=-8

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Example Question #1 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 9&3 \\21 &11 \end{vmatrix}$

Possible Answers:

249

162

156

213

Correct answer:

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

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

For the matrix $A=\begin{vmatrix} 9&3 \\21 &11 \end{vmatrix}$

The determinant is thus:

|A|=99-63=36

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Example Question #10 : Matrices

Find the determinant of the matrix $A=\begin{vmatrix} 11&3 \\2 &4 \end{vmatrix}$

Possible Answers:

Correct answer:

Explanation:

The determinant of a 2x2 matrix can be found by cross multiplying terms as follows:

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

For the matrix $A=\begin{vmatrix} 11&3 \\2 &4 \end{vmatrix}$

The determinant is thus:

|A|=44-6=38

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Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #1 : Matrices

Example Question #1 : Matrices

Example Question #2 : Matrices

Example Question #1 : Matrices

Example Question #2 : Matrices

Example Question #1 : Matrices

Example Question #7 : Matrices

Example Question #8 : Matrices

Example Question #1 : Matrices

Example Question #10 : Matrices

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