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% Document Info
\author{Trina Dey}
\pdfinfo{
  /Title (matrices.pdf)
  /Creator (Cheatography)
  /Author (Trina Dey)
  /Subject (Matrices Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Matrices Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Trina Dey} via \textcolor{DarkBackground}{\uline{cheatography.com/136953/cs/28643/}}}
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\noindent
\begin{multicols}{3}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Trina Dey \\
  \uline{cheatography.com/trina-dey} \\
  \end{tabulary}
\vfill
\columnbreak
\begin{tabulary}{5.8cm}{L}
  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Published 3rd September, 2021.\\
   Updated 3rd September, 2021.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Sponsor}}  \\
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  \vspace{-5pt}
  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{document}
\raggedright
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{1.69218 cm} x{3.28482 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Matrices}}  \tn
% Row 0
\SetRowColor{LightBackground}
Addition & X + Y = {[}zij{]} = {[}xij + yij{]} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Subtraction & X - Y = {[}zij{]} = {[}xij - yij{]} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\seqsplit{Multiplication} & X * Y = {[}zij{]} = {[} xi * yj{]} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
Constant & c * X = {[}zij{]} = {[}c * xij{]} \tn 
% Row Count 7 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.58804 cm} x{2.38896 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Transpose \& Identity}}  \tn
% Row 0
\SetRowColor{LightBackground}
Transpose & X\textasciicircum{}T\textasciicircum{} = {[}zij{]} = {[}xji{]} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
Tr of Tr & (X\textasciicircum{}T\textasciicircum{})\textasciicircum{}T\textasciicircum{} = X \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
Tr of Mul & (XY)\textasciicircum{}T\textasciicircum{} = Y\textasciicircum{}T\textasciicircum{} {\emph{ X\textasciicircum{}T\textasciicircum{} != X\textasciicircum{}T\textasciicircum{} }} Y\textasciicircum{}T\textasciicircum{} \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
Sym Matrix & X\textasciicircum{}T\textasciicircum{} = X \tn 
% Row Count 6 (+ 1)
% Row 4
\SetRowColor{LightBackground}
Identity Matrix I \{\{nl\}\} {[}zii=1, zij=0{]} & X {\emph{ I = I }} X = X \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.09494 cm} x{3.88206 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Inverse}}  \tn
% Row 0
\SetRowColor{LightBackground}
Inverse & X X\textasciicircum{}-1\textasciicircum{} = I = X\textasciicircum{}-1\textasciicircum{}X \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{if X\textasciicircum{}-1\textasciicircum{} exists then X is non singular or invertible} \tn 
% Row Count 3 (+ 2)
% Row 2
\SetRowColor{LightBackground}
Inv of Inv & (X\textasciicircum{}-1\textasciicircum{})\textasciicircum{}-1\textasciicircum{} = X \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
Inv of Mul & (XY)\textasciicircum{}-1\textasciicircum{} = Y\textasciicircum{}-1\textasciicircum{}X\textasciicircum{}-1\textasciicircum{} != X\textasciicircum{}-1\textasciicircum{}Y\textasciicircum{}-1\textasciicircum{} \tn 
% Row Count 7 (+ 2)
% Row 4
\SetRowColor{LightBackground}
Inv of Tr & (X\textasciicircum{}T\textasciicircum{})\textasciicircum{}-1\textasciicircum{} = (X\textasciicircum{}-1\textasciicircum{})\textasciicircum{}T\textasciicircum{} \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
\seqsplit{Determinant} & |A| = \textasciicircum{}n\textasciicircum{}∑`i=1`  a`i``j` x Det |a`i``j`| \tn 
% Row Count 11 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{2}{x{5.377cm}}{Determinant is computed over first row of matrix where each element of first row is multiplied by its minor} \tn 
% Row Count 14 (+ 3)
% Row 7
\SetRowColor{white}
\mymulticolumn{2}{x{5.377cm}}{minor M`i``j` is a determinant obtained by deleting the i\textasciicircum{}th\textasciicircum{} row and j\textasciicircum{}th\textasciicircum{} column in which a`i``j` lies. Minor of a`i``j` is denoted by m`i``j`.} \tn 
% Row Count 17 (+ 3)
% Row 8
\SetRowColor{LightBackground}
\seqsplit{Cofactor} & A`i``j` = (-1)\textasciicircum{}i+j\textasciicircum{} m`i``j` \tn 
% Row Count 18 (+ 1)
% Row 9
\SetRowColor{white}
Adjoint & adj(A) = (Cofactor)\textasciicircum{}T\textasciicircum{} = (A`i``j`)\textasciicircum{}T\textasciicircum{} \tn 
% Row Count 20 (+ 2)
% Row 10
\SetRowColor{LightBackground}
Inverse & A\textasciicircum{}-1\textasciicircum{} = adj(A) / |A| \tn 
% Row Count 21 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Orthogonal}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Two n x 1 vectors are orthogonal if  X\textasciicircum{}T\textasciicircum{} Y = 0} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{A vector is orthonormal if X\textasciicircum{}T\textasciicircum{}X = ||X\textasciicircum{}2\textasciicircum{}||} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Sq root of ||X|| is length or norm of vector} \tn 
% Row Count 3 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{\{X`1`, X`2`, X`3`.... X`n`) are said to be orthonormal if, each pair is orthogonal and have unit length} \tn 
% Row Count 6 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{A sq matrix is orthogonal if X\textasciicircum{}T\textasciicircum{}X= I or X\textasciicircum{}T\textasciicircum{}=X\textasciicircum{}-1\textasciicircum{}} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Eigen Values \& Eigen Vectors}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{A is nxn matrix, X is nx1 matrix, λ is a scalar, then} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{AX = λX or (A-λI)X = 0 or X = (A-λI)\textasciicircum{}-1\textasciicircum{}} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{λ is the eigen value and X is the eigen vector (non zero)} \tn 
% Row Count 5 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Since X is non zero, |A-λI| should be 0} \tn 
% Row Count 6 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Determinant for {[}a b{]} = ad - bc \{\{nl\}\}~~~~~~~~~~~~~~~~~~~~~~~~~{[}c d{]}} \tn 
% Row Count 10 (+ 4)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If A =\textgreater{} symmetric, then eigenvalues =\textgreater{} real \& \{\{nl\}\} eigenvectors =\textgreater{} orthogonal} \tn 
% Row Count 12 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Diagonalization: P =\textgreater{} orthogonal matrix, then Z = P\textasciicircum{}T\textasciicircum{}AP, Z is diagonal matrix with eigen values of A} \tn 
% Row Count 15 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Linear Independence}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Given a`1`x`1` + a`2`x`2` + ...a`n`x`n` = 0, if a vector  {[}a`1`, a`2`, ...a`n`{]} exists such that} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{a. all a`i` are 0, then x`i` are linearly independent.} \tn 
% Row Count 4 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{b. if some a`i`!=0 then x`i` are linearly dependent.} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If a set of vectors are linearly dependent, then one of them can be written as some combination of others} \tn 
% Row Count 9 (+ 3)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{A set of two vectors is linearly dependent if and only if one of the vectors is a constant multiple of the other.} \tn 
% Row Count 12 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Idempotence}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{a nxn matrix A is idempotent iff A\textasciicircum{}2\textasciicircum{} = A} \tn 
% Row Count 1 (+ 1)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{The identity matrix I is idempotent.} \tn 
% Row Count 2 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Let X be an n×k matrix of full rank ,n≥k then H exists as H=X(X\textasciicircum{}T\textasciicircum{}X)\textasciicircum{}−1\textasciicircum{}X\textasciicircum{}T\textasciicircum{} and is idempotent.} \tn 
% Row Count 4 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{ColBreak2}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Rank}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{For a nxk matrix say X, the column vectors are {[}x`1`, x`2`, ...x`k`{]} and {\bf{{\emph{rank}}}} is given by max num of linearly independent vectors.} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If X is a nxk matrix and r(X) = k, then X is of full rank for n≥k.} \tn 
% Row Count 5 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{r(X) = r(X\textasciicircum{}T\textasciicircum{}) = r(X\textasciicircum{}T\textasciicircum{}X)} \tn 
% Row Count 6 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If X is kxk, then X is non singular iff r(X) = k.} \tn 
% Row Count 7 (+ 1)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{If X is n×k, P is n×n and non-singular, and Q is k×k and nonsingular, then r(X) =r(PX) =r(XQ).} \tn 
% Row Count 9 (+ 2)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{The rank of a diagonal matrix is equal to the number of non zero diagonal entries in the matrix.} \tn 
% Row Count 11 (+ 2)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{r(XY) ≤ r(X) r(Y)} \tn 
% Row Count 12 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Trace}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{The trace of a square k×k matrix X is sum of its diagonal entries - \{\{nl\}\} tr(X) = ∑ xii} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If c is a scalar, tr(cX) =c * tr(X)} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{tr(X±Y) =tr(X) ± tr(Y).} \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If XY and YX both exist, tr(XY) =tr(YX).} \tn 
% Row Count 5 (+ 1)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Quadratic Forms}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{A be a k × k, y be k × 1 vector containing variables q = y\textasciicircum{}T\textasciicircum{} Ay is called a quadratic form in y, A is called the matrix of the quadratic form} \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{q = ∑ ∑ a`ij`y`i`y`j`} \tn 
% Row Count 4 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{If y\textasciicircum{}T\textasciicircum{}Ay \textgreater{} 0 for all y != 0, y\textasciicircum{}T\textasciicircum{}Ay \& A are +ve definite} \tn 
% Row Count 6 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{If y\textasciicircum{}T\textasciicircum{}Ay \textgreater{}= 0 for all y != 0, y\textasciicircum{}T\textasciicircum{}Ay \& A are +ve semidefinite} \tn 
% Row Count 8 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Matrix Differentiation}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{y = (y1, y2, . . . , yk)\textasciicircum{}T\textasciicircum{},  z = f(y) then ∂z/∂y = {[}∂z/∂y`1` ∂z/∂y`2` ∂z/∂y`3`{]}\textasciicircum{}T\textasciicircum{}} \tn 
% Row Count 2 (+ 2)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{z=a\textasciicircum{}T\textasciicircum{}y,  ∂z/∂y = a} \tn 
% Row Count 3 (+ 1)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{z=y\textasciicircum{}T\textasciicircum{}y,  ∂z/∂y = 2y} \tn 
% Row Count 4 (+ 1)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{z=y\textasciicircum{}T\textasciicircum{}Ay,  ∂z/∂y = Ay+A\textasciicircum{}T\textasciicircum{}y , if A is symmetrix then ∂z/∂y = 2Ay} \tn 
% Row Count 6 (+ 2)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Page Break1}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{} \tn 
% Row Count 0 (+ 0)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{X}
\SetRowColor{DarkBackground}
\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Theorems}}  \tn
% Row 0
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Theorem 1\{\{nl\}\} Let A be a symmetric k×k  matrix.  Then an orthogonal matrix P exists such that P\textasciicircum{}T\textasciicircum{}AP = λ x I, where λ = {[}λ`1`, λ`2`, .... λ`n` {]} are the eigen values of A as nx1 vector} \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Theorem 2\{\{nl\}\} The eigenvalues of idempotent matrices are always either 0 or 1.} \tn 
% Row Count 6 (+ 2)
% Row 2
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Theorem 3\{\{nl\}\} If A is a symmetric and idempotent matrix, r(A) =tr(A)} \tn 
% Row Count 8 (+ 2)
% Row 3
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Theorem 4\{\{nl\}\} Let A`1`,A`2`,...,A`m` be a collection of symmetric k×k  matrices. Then the following are equivalent: \{\{nl\}\}a.  There exists an orthogonal matrix P  such that P\textasciicircum{}T\textasciicircum{}A`i`P  is diagonal for all i= 1,2,...,m; \{\{nl\}\} b.  A`i`A`j`=A`j`A`i` for every pair i,j= 1,2,...,m.} \tn 
% Row Count 14 (+ 6)
% Row 4
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Theorem 5\{\{nl\}\} Let A`1`,A`2`,...,A`m` be a collection of symmetric k×k  matrices. Then any two of the following conditions implies the third: \{\{nl\}\} a. All Ai, i= 1,2,...,m are idempotent; \{\{nl\}\} b. ∑ Ai is idempotent; \{\{nl\}\} c. AiAj= 0for i6=j} \tn 
% Row Count 19 (+ 5)
% Row 5
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Theorem 6 \{\{nl\}\} Let A`1`,A`2`,...,A`m` be a collection of symmetric k×k  matrices.  If the conditions in Theorem 5 are true, then \{\{nl\}\} r(∑A`i`) = ∑r(A`i`)} \tn 
% Row Count 23 (+ 4)
% Row 6
\SetRowColor{LightBackground}
\mymulticolumn{1}{x{5.377cm}}{Theroem 7 \{\{nl\}\} A symmetric matrix A is positive definite if and only if its eigen values are all (strictly) positive} \tn 
% Row Count 26 (+ 3)
% Row 7
\SetRowColor{white}
\mymulticolumn{1}{x{5.377cm}}{Theorem 8 \{\{nl\}\} A symmetric matrix A is positive semi-definite if and only if its eigenvalues are all non-negative.} \tn 
% Row Count 29 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}-}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}