Matrix multiplication with transpose

Problem

We have \(x_1={\left( 1,-1 \right)}'\) , \(x_2={\left( 0,2 \right)}'\) and \(x_3={\left( -2,0 \right)}'\)

a) Create a \(3×2\) matrix \(X\) as

\[X=\begin{bmatrix}x'_1 \\ x'_2 \\ x'_3\end{bmatrix}\]

b) Calculate

\[\sum_{i=1}^{3}{ x_ix'_i }\]

c) Calculate

\[X'X\]

• and convince yourself that

\[X'X=\sum_{i=1}^{3}{ x_ix'_i }\]

• d) Now let \(x_1={\left( x_{11},x_{12} \right)}'\) , \(x_2={\left( x_{21},x_{22} \right)}'\) and \(x_3={\left( x_{31},x_{32} \right)}'\) . Then redo parts a) through c)