Exercise 1.1.17

Question

For these matrices with square blocks, find $A=C R$. What ranks?
$$
A_{1}=\left[\begin{array}{cc}
	ext { zeros } & 	ext { ones } \
	ext { ones } & 	ext { ones }
\end{array}ight]_{\mathbf{4} 	imes \mathbf{4}} \quad A_{2}=\left[\begin{array}{c}
A_{1} \
A_{1}
\end{array}ight]_{\mathbf{8} 	imes \mathbf{4}} \quad A_{3}=\left[\begin{array}{cc}
A_{1} & A_{1} \
A_{1} & A_{1}
\end{array}ight]_{\mathbf{8} 	imes \mathbf{8}}
$$

Neil Z. · Accepted Answer

\begin{itemize}
\item
  $A_1 = C_1R_1 = \begin{bmatrix}0 & 1\0 & 1 \ 1 & 1 \ 1 & 1\end{bmatrix}\begin{bmatrix}1 & 1 & 0 & 0 \ 0 & 0 & 1 & 1\end{bmatrix}$
\item
  $A_2 = C_2R_2 = \begin{bmatrix}A_1 \ A_1 \end{bmatrix} = \begin{bmatrix}C_1R_1 \ C_1R_1 \end{bmatrix} = \begin{bmatrix}C_1 \ C_1 \end{bmatrix}R_1$.
  So $C_2 = \begin{bmatrix}C_1 \ C_1 \end{bmatrix}$ and $R_2=R_1$.
\item
  $A_3 = C_3R_3 = \begin{bmatrix}A_1 &A_1 \ A_1 & A_1 \end{bmatrix} = \begin{bmatrix}C_1R_1 & C_1R_1 \ C_1R_1 & C_1R_1\end{bmatrix} = \begin{bmatrix}C_1 \ C_1 \end{bmatrix}\begin{bmatrix}R_1 & R_1\end{bmatrix}$
  So $C_3 = \begin{bmatrix}C_1 \ C_1 \end{bmatrix}$ and
  $R_3 = \begin{bmatrix}R_1 & R_1\end{bmatrix}$
\end{itemize}

All matrices have $rank=2$

Exercise 1.1.17

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Exercise 1.1.17

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