Calculus 3 : Matrices

Study concepts, example questions & explanations for Calculus 3

varsity tutors app store varsity tutors android store

Example Questions

Example Question #1 : Matrices

Calculate the determinant of Matrix \(\displaystyle A\).

\(\displaystyle A=\begin{bmatrix} 1 & 2&3 \\ 0& 2& 1\\ 3& 4& 5 \end{bmatrix}\)

Possible Answers:

\(\displaystyle \det(A)=-6\)

\(\displaystyle \det(A)=1\)

\(\displaystyle \det(A)=-2\)

\(\displaystyle \det(A)=0\)

\(\displaystyle \det(A)=6\)

Correct answer:

\(\displaystyle \det(A)=-6\)

Explanation:

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

\(\displaystyle A=\begin{bmatrix} 1 & 2&3 & 1 &2\\ 0& 2& 1&0 &2\\ 3& 4& 5 &3 &4\end{bmatrix}\)

The next step is to multiply the down diagonals. 

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

The next step is to multiply the up diagonals.

\(\displaystyle A_u=3\cdot2\cdot3+4\cdot1\cdot1+5\cdot0\cdot2=18+4=22\)

The last step is to substract \(\displaystyle A_u\) from \(\displaystyle A_d\).

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

 

Example Question #1 : Matrices

Calculate the determinant of Matrix \(\displaystyle A\).

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

Possible Answers:

\(\displaystyle \det(A)=-38\)

\(\displaystyle \det(A)=0\)

\(\displaystyle \det(A)=38\)

\(\displaystyle \det(A)=67\)

\(\displaystyle \det(A)=167\)

Correct answer:

\(\displaystyle \det(A)=167\)

Explanation:

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

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

The next step is to multiply the down diagonals. 

\(\displaystyle A_d=0\cdot-3\cdot-5+1\cdot2\cdot 1+5\cdot10\cdot2=0+2+100=102\)

The next step is to multiply the up diagonals.

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

The last step is to substract \(\displaystyle A_u\) from \(\displaystyle A_d\).

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

Example Question #1 : Matrices

Calculate the determinant of \(\displaystyle A\).

\(\displaystyle A=\begin{bmatrix} 3 & 4\\ -2 & 10 \end{bmatrix}\)

Possible Answers:

\(\displaystyle 22\)

\(\displaystyle 2\)

\(\displaystyle 38\)

\(\displaystyle 50\)

\(\displaystyle 0\)

Correct answer:

\(\displaystyle 38\)

Explanation:

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

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

Example Question #1 : Matrices

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

Possible Answers:

\(\displaystyle \begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5& 2 & 4 \end{bmatrix}\)

All of the above answers

\(\displaystyle \begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5\\ 2\\ 4\\ \end{bmatrix}\)

None of the other answers

\(\displaystyle \begin{bmatrix} 2 \\ 6 \\ 1 \end{bmatrix} \begin{bmatrix} 5 \\ 2 \\ 4 \end{bmatrix}\)

Correct answer:

\(\displaystyle \begin{bmatrix} 2 & 6 & 1 \end{bmatrix} \begin{bmatrix} 5\\ 2\\ 4\\ \end{bmatrix}\)

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

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

Which is the same as

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

 

This operation is true for (real) \(\displaystyle 3\)-dimensional vectors in general;

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

Example Question #2 : Matrices

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

Possible Answers:

\(\displaystyle 19\)

\(\displaystyle 1\)

\(\displaystyle 15\)

\(\displaystyle -19\)

\(\displaystyle -1\)

Correct answer:

\(\displaystyle -1\)

Explanation:

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

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

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

The determinant is thus:

\(\displaystyle |A|=9-10=-1\)

Example Question #1 : Matrices

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

Possible Answers:

\(\displaystyle 12\)

\(\displaystyle 10\)

\(\displaystyle 5\)

\(\displaystyle 11\)

\(\displaystyle 14\)

Correct answer:

\(\displaystyle 14\)

Explanation:

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

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

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

The determinant is thus:

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

Example Question #7 : Matrices

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

Possible Answers:

\(\displaystyle 11\)

\(\displaystyle 13\)

\(\displaystyle 17\)

\(\displaystyle 7\)

\(\displaystyle -1\)

Correct answer:

\(\displaystyle -1\)

Explanation:

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

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

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

The determinant is thus:

\(\displaystyle |A|=5-6=-1\)

Example Question #8 : Matrices

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

Possible Answers:

\(\displaystyle 0\)

\(\displaystyle -12\)

\(\displaystyle 8\)

\(\displaystyle 16\)

\(\displaystyle -8\)

Correct answer:

\(\displaystyle -8\)

Explanation:

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

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

For the matrix \(\displaystyle A=\begin{vmatrix} 4&4 \\4 &2 \end{vmatrix}\)

The determinant is thus:

\(\displaystyle |A|=8-16=-8\)

Example Question #1 : Matrices

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

Possible Answers:

\(\displaystyle 249\)

\(\displaystyle 36\)

\(\displaystyle 162\)

\(\displaystyle 156\)

\(\displaystyle 213\)

Correct answer:

\(\displaystyle 36\)

Explanation:

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

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

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

The determinant is thus:

\(\displaystyle |A|=99-63=36\)

Example Question #10 : Matrices

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

Possible Answers:

\(\displaystyle 34\)

\(\displaystyle 10\)

\(\displaystyle 50\)

\(\displaystyle 25\)

\(\displaystyle 38\)

Correct answer:

\(\displaystyle 38\)

Explanation:

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

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

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

The determinant is thus:

\(\displaystyle |A|=44-6=38\)

Learning Tools by Varsity Tutors