The Trace - Linear Algebra

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Question

Calculate the trace of the following Matrix.

$\begin{bmatrix} 2&4 &45 \ 3& 10& 30\ 100& 3& 56 \end{bmatrix}$

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Answer

The trace of a matrix is simply adding the entries along the main diagonal.

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Question

Calculate the trace.

$\begin{bmatrix} 1&4 \ 5&-1 \end{bmatrix}$

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Answer

The trace of a matrix is simply adding the entries along the main diagonal.

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Question

Find the trace of the following matrix.

$\begin{bmatrix} 14 & 15 \ 2 & -5\ 5& 10 \end{bmatrix}$

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Answer

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

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Question

Calculate the trace of the following matrix.

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

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Answer

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

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Question

Calculate the trace of matrix , given

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

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Answer

By definition,

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

Therefore,

.

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Question

Calculate the trace of , or , given

$A= \begin{bmatrix} 1&2 &5 \ 4&7 &1 \end{bmatrix}$ .

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Answer

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

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Question

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

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Answer

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

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Question

$\begin{align}&\text{Compute }Tr\begin{bmatrix}-10&-7\7&-15\end{bmatrix}\end{align}$

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Answer

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

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Question

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

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Answer

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

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Question

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

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Answer

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

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Question

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

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Answer

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

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Question

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

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Answer

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

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Question

$\begin{align}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}1&6&-4&13\9&19&1&-7\-16&5&11&-3\-17&-10&-14&-9\end{bmatrix}\end{align}$

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Answer

$\begin{align}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\&\text{i.e. }Tr\begin{bmatrix}a&b&c\d&e&f\g&h&i\end{bmatrix}=a+e+i\&\text{For the matrix }A=\begin{bmatrix}1&6&-4&13\9&19&1&-7\-16&5&11&-3\-17&-10&-14&-9\end{bmatrix}\&\text{We find: }\&Tr(A)=(1)+(19)+(11)+(-9)=22\end{align}$

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Question

$\begin{align}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}6&19&-11&7&-9&7&8&-18\-10&-11&7&14&-6&12&7&-20\4&-5&17&-20&-2&-3&-2&11\-7&12&-1&-19&-13&9&-1&-14\-7&4&-13&10&-11&17&-9&11\-13&-9&-17&3&8&2&-3&6\6&7&6&18&-12&9&-11&-16\4&-2&-2&7&11&-6&7&-3\end{bmatrix}\end{align}$

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Answer

$\begin{align}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\&\text{i.e. }Tr\begin{bmatrix}a&b&c\d&e&f\g&h&i\end{bmatrix}=a+e+i\&\text{For the matrix }A=\begin{bmatrix}6&19&-11&7&-9&7&8&-18\-10&-11&7&14&-6&12&7&-20\4&-5&17&-20&-2&-3&-2&11\-7&12&-1&-19&-13&9&-1&-14\-7&4&-13&10&-11&17&-9&11\-13&-9&-17&3&8&2&-3&6\6&7&6&18&-12&9&-11&-16\4&-2&-2&7&11&-6&7&-3\end{bmatrix}\&\text{We find: }\&Tr(A)=(6)+(-11)+(17)+(-19)+(-11)+(2)+(-11)+(-3)=-30\end{align}$

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Question

$\begin{align}&\text{Calculate the trace of the matrix}\&\begin{bmatrix}15&-10&-7&-16&18\6&-1&6&2&6\2&9&1&20&-12\-16&-16&-18&-4&-2\-6&11&5&11&18\end{bmatrix}\end{align}$

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Answer

$\begin{align}&\text{The trace of a square nxn matrix is the sum of its diagonal elements:}\&\sum_{i=1}^{n}A_{i,i}\&\text{For the matrix }\begin{bmatrix}15&-10&-7&-16&18\6&-1&6&2&6\2&9&1&20&-12\-16&-16&-18&-4&-2\-6&11&5&11&18\end{bmatrix}\&\text{This becomes: }\&(15)+(-1)+(1)+(-4)+(18)=29\end{align}$

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Question

$\begin{align}&\text{Compute }Tr\begin{bmatrix}1&15&-1&-4&7\10&1&-6&-14&4\-10&-19&10&-11&-2\8&-6&10&-4&8\8&-2&-20&-7&-3\end{bmatrix}\end{align}$

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Answer

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

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Question

$\begin{align}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}13&10&-5&-12\12&18&-7&7\-3&14&11&-14\15&20&1&16\end{bmatrix}\end{align}$

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Answer

$\begin{align}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\&\text{i.e. }Tr\begin{bmatrix}a&b&c\d&e&f\g&h&i\end{bmatrix}=a+e+i\&\text{For the matrix }A=\begin{bmatrix}13&10&-5&-12\12&18&-7&7\-3&14&11&-14\15&20&1&16\end{bmatrix}\&\text{We find: }\&Tr(A)=(13)+(18)+(11)+(16)=58\end{align}$

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Question

$\begin{align}&\text{Find }Tr(A)\text{ for }A=\begin{bmatrix}12&-7&1&-17&-16&-15&7\0&-13&0&-14&-18&14&2\18&8&3&13&16&20&-20\15&5&20&1&-1&12&-11\0&16&3&14&10&4&-10\7&-17&5&7&9&16&20\11&3&18&3&-20&-16&15\end{bmatrix}\end{align}$

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Answer

$\begin{align}&\text{Tr, that is to say the trace, of a square nxn matrix is the its diagonal elements:}\&\text{i.e. }Tr\begin{bmatrix}a&b&c\d&e&f\g&h&i\end{bmatrix}=a+e+i\&\text{For the matrix }A=\begin{bmatrix}12&-7&1&-17&-16&-15&7\0&-13&0&-14&-18&14&2\18&8&3&13&16&20&-20\15&5&20&1&-1&12&-11\0&16&3&14&10&4&-10\7&-17&5&7&9&16&20\11&3&18&3&-20&-16&15\end{bmatrix}\&\text{We find: }\&Tr(A)=(12)+(-13)+(3)+(1)+(10)+(16)+(15)=44\end{align}$

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Question

$A = \begin{bmatrix} 3&0 & 0\ 0& 6& 0\ 0&0 &x \end{bmatrix}$ ,

where is a complex number. The trace of $A^{2}$ is 50.

Evaluate .

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Answer

is a diagonal matrix - its only nonzero elements are on the main diagonal, from upper left to lower right - so its square can be taken by simply squaring the diagonal elements. Since

$A = \begin{bmatrix} 3&0 & 0\ 0& 6& 0\ 0&0 &x \end{bmatrix}$

it follows that

$A ^{2}= \begin{bmatrix} 3^{2}&0 & 0\ 0& 6^{2}& 0\ 0&0 &x^{2} \end{bmatrix} = \begin{bmatrix} 9&0 & 0\ 0& 36 & 0\ 0&0 &x^{2} \end{bmatrix}$

The trace of a matrix is equal to the sum of the elements in its main diagonal, so the trace of $A^{2}$ is

$Tr(A^{2}) = 9 + 36 + x^{2}$

$Tr(A^{2}) = x^{2} + 45$

Since the trace is given to be 50, set this equal to 50 and solve for :

$x^{2} + 45= 50$

$x^{2} =5$

$x= \pm \sqrt{5}$

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Question

All skew-symmetric matrices have a trace of

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Answer

For any skew-symmetric matrix , . Taking the trace of both sides we get

.

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