Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

varsity tutors app store varsity tutors android store

Example Questions

Example Question #103 : Operations And Properties

Find the trace of the following matrix.

\(\displaystyle \begin{bmatrix} 14 & 15 \\ 2 & -5\\ 5& 10 \end{bmatrix}\)

Possible Answers:

\(\displaystyle 17\)

\(\displaystyle 16\)

Not possible to calculate

\(\displaystyle 9\)

\(\displaystyle 10\)

Correct answer:

Not possible to calculate

Explanation:

Since the trace can only be calculated for \(\displaystyle n\times n\) matrices, the trace isn't possible to calculate.

Example Question #104 : Operations And Properties

Calculate the trace of the following matrix.

\(\displaystyle \begin{bmatrix} -3& 10&11 \\ 3& 25& 20\\ 6& -45& -22 \end{bmatrix}\)

Possible Answers:

\(\displaystyle Trace=0\)

\(\displaystyle Trace=-61\)

\(\displaystyle Trace=18\)

\(\displaystyle Trace=48\)

\(\displaystyle Trace=-10\)

Correct answer:

\(\displaystyle Trace=0\)

Explanation:

In order to calculate the trace, we need to sum up each entry along the main diagonal.

\(\displaystyle Trace=-3+25-22=0\)

Example Question #105 : Operations And Properties

Calculate the trace of matrix \(\displaystyle A\), given

 \(\displaystyle A=\begin{bmatrix} 0& 6& 8&22 \\ -13& 0&32 &25 \\ 11& 0&-3 &43 \\ 4&31 &98 &4 \end{bmatrix}\).

Possible Answers:

\(\displaystyle tr(A)= 1\)

\(\displaystyle tr(A)= 33\)

\(\displaystyle tr(A)= -1\)

\(\displaystyle tr(A)= 73\)

Correct answer:

\(\displaystyle tr(A)= 1\)

Explanation:

By definition, 

\(\displaystyle tr(A)= \sum_{i=1}^n a_{ii}\).

Therefore, 

\(\displaystyle tr(A)= 0+0+(-3)+4=1\).

Example Question #106 : Operations And Properties

Calculate the trace of \(\displaystyle A\), or \(\displaystyle tr(A)\), given 

\(\displaystyle A= \begin{bmatrix} 1&2 &5 \\ 4&7 &1 \end{bmatrix}\).

Possible Answers:

\(\displaystyle tr(A) = 3\)

\(\displaystyle tr(A) = 8\)

\(\displaystyle tr(A) = 0\)

The trace does not exist.

\(\displaystyle tr(A) = 13\)

Correct answer:

The trace does not exist.

Explanation:

By definition, the trace of a matrix only exists in the matrix is a square matrix.  In this case, \(\displaystyle A\) is not square.  Therefore, the trace does not exist.

Example Question #107 : Operations And Properties

\(\displaystyle \begin{align*}&\text{Determine the trace of the matrix}\\&A=\begin{bmatrix}-8&-7\\-3&0\end{bmatrix}\end{align*}\)

Possible Answers:

\(\displaystyle -2\)

\(\displaystyle -7\)

\(\displaystyle -17\)

\(\displaystyle -8\)

Correct answer:

\(\displaystyle -8\)

Explanation:

\(\displaystyle \begin{align*}&\text{We can find the trace of a square nxn matrix by summing its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For }A=\begin{bmatrix}-8&-7\\-3&0\end{bmatrix}\\&(-8)+(0)=-8\end{align*}\)

Example Question #108 : Operations And Properties

\(\displaystyle \begin{align*}&\text{Compute }Tr\begin{bmatrix}-10&-7\\7&-15\end{bmatrix}\end{align*}\)

Possible Answers:

\(\displaystyle -25\)

\(\displaystyle -22\)

\(\displaystyle -20\)

\(\displaystyle -34\)

Correct answer:

\(\displaystyle -25\)

Explanation:

\(\displaystyle \begin{align*}&\text{Tr in the context of linear algebra represents the trace.}\\&\text{We can find this quantity for a square nxn matrix by summing the elements of its diagonal:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{Doing so we find for}\begin{bmatrix}-10&-7\\7&-15\end{bmatrix}\\&\text{We find:}\\&(-10)+(-15)=-25\end{align*}\)

Example Question #187 : Linear Algebra

\(\displaystyle \begin{align*}&\text{Compute }Tr\begin{bmatrix}-7&8&-12&-19&10&0&-1&17\\5&5&15&13&3&-13&-11&16\\-19&0&-14&20&9&0&-1&-18\\7&-19&-18&1&-17&13&13&9\\-14&7&1&19&6&12&-2&-3\\13&-17&-15&-13&-4&14&12&-18\\-4&1&-3&6&5&-9&-3&-20\\20&-14&-16&-5&-12&0&-7&19\end{bmatrix}\end{align*}\)

Possible Answers:

\(\displaystyle 21\)

\(\displaystyle 13\)

\(\displaystyle 27\)

\(\displaystyle 24\)

Correct answer:

\(\displaystyle 21\)

Explanation:

\(\displaystyle \begin{align*}&\text{Tr in the context of linear algebra represents the trace.}\\&\text{We can find this quantity for a square nxn matrix by summing the elements of its diagonal:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{Doing so we find for}\begin{bmatrix}-7&8&-12&-19&10&0&-1&17\\5&5&15&13&3&-13&-11&16\\-19&0&-14&20&9&0&-1&-18\\7&-19&-18&1&-17&13&13&9\\-14&7&1&19&6&12&-2&-3\\13&-17&-15&-13&-4&14&12&-18\\-4&1&-3&6&5&-9&-3&-20\\20&-14&-16&-5&-12&0&-7&19\end{bmatrix}\\&\text{We find:}\\&(-7)+(5)+(-14)+(1)+(6)+(14)+(-3)+(19)=21\end{align*}\)

Example Question #109 : Operations And Properties

\(\displaystyle \begin{align*}&\text{Determine the trace of the matrix}\\&A=\begin{bmatrix}20&-8&8&7\\2&8&7&-13\\-15&20&-13&-19\\3&16&7&-13\end{bmatrix}\end{align*}\)

Possible Answers:

\(\displaystyle 8\)

\(\displaystyle -7\)

\(\displaystyle 4\)

\(\displaystyle 2\)

Correct answer:

\(\displaystyle 2\)

Explanation:

\(\displaystyle \begin{align*}&\text{We can find the trace of a square nxn matrix by summing its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For }A=\begin{bmatrix}20&-8&8&7\\2&8&7&-13\\-15&20&-13&-19\\3&16&7&-13\end{bmatrix}\\&Tr(A)=(20)+(8)+(-13)+(-13)=2\end{align*}\)

Example Question #11 : The Trace

\(\displaystyle \begin{align*}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}-3&-1&-16\\4&-11&-5\\3&-10&-9\end{bmatrix}\end{align*}\)

Possible Answers:

\(\displaystyle -29\)

\(\displaystyle -24\)

\(\displaystyle -23\)

\(\displaystyle -21\)

Correct answer:

\(\displaystyle -23\)

Explanation:

\(\displaystyle \begin{align*}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\\&\text{i.e. }Tr\begin{bmatrix}a&b&c\\d&e&f\\g&h&i\end{bmatrix}=a+e+i\\&\text{For the matrix }A=\begin{bmatrix}-3&-1&-16\\4&-11&-5\\3&-10&-9\end{bmatrix}\\&\text{We find: }\\&Tr(A)=(-3)+(-11)+(-9)=-23\end{align*}\)

Example Question #12 : The Trace

\(\displaystyle \begin{align*}&\text{Calculate the trace of the matrix}\\&\begin{bmatrix}3&-16&17&16\\13&-10&4&-20\\-3&-8&-14&-13\\-3&-17&4&-1\end{bmatrix}\end{align*}\)

Possible Answers:

\(\displaystyle -17\)

\(\displaystyle -22\)

\(\displaystyle -25\)

\(\displaystyle 9\)

Correct answer:

\(\displaystyle -22\)

Explanation:

\(\displaystyle \begin{align*}&\text{The trace of a square nxn matrix is the sum of its diagonal elements:}\\&\sum_{i=1}^{n}A_{i,i}\\&\text{For the matrix }\begin{bmatrix}3&-16&17&16\\13&-10&4&-20\\-3&-8&-14&-13\\-3&-17&4&-1\end{bmatrix}\\&\text{This becomes: }\\&(3)+(-10)+(-14)+(-1)=-22\end{align*}\)

Learning Tools by Varsity Tutors