diff --git a/docs/src/padarrays.md b/docs/src/padarrays.md index ea1bd58..1952a04 100644 --- a/docs/src/padarrays.md +++ b/docs/src/padarrays.md @@ -46,6 +46,7 @@ The following examples show the effects of modifying the input array: ```math \mathbf{A} = \boxed{ +\textcolor{red}{ \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 4 & 6 & 8 & 10 & 12 \\ @@ -53,7 +54,7 @@ The following examples show the effects of modifying the input array: 4 & 8 & 12 & 16 & 20 & 24 \\ 5 & 10 & 15 & 20 & 25 & 30 \\ 6 & 12 & 18 & 24 & 30 & 36 - \end{matrix}}. + \end{matrix}}}. ``` ### Examples with `Pad` @@ -67,12 +68,12 @@ The command `padarray(A, Pad(:replicate, 4, 4))` yields 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 \\ 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 \\ 1 & 1 & 1 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 6 & 6 & 6 \\ -1 & 1 & 1 & 1 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 6 & 6 & 6 & 6 \\ -2 & 2 & 2 & 2 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 12 & 12 & 12 & 12 \\ -3 & 3 & 3 & 3 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 18 & 18 & 18 & 18 \\ -4 & 4 & 4 & 4 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 24 & 24 & 24 & 24 \\ -5 & 5 & 5 & 5 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 30 & 30 & 30 & 30 \\ -6 & 6 & 6 & 6 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 36 & 36 & 36 & 36 \\ +1 & 1 & 1 & 1 & \textcolor{red}{1} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{6} & 6 & 6 & 6 & 6 \\ +2 & 2 & 2 & 2 & \textcolor{red}{2} & \textcolor{red}{4} & \textcolor{red}{6} & \textcolor{red}{8} & \textcolor{red}{10} & \textcolor{red}{12} & 12 & 12 & 12 & 12 \\ +3 & 3 & 3 & 3 & \textcolor{red}{3} & \textcolor{red}{6} & \textcolor{red}{9} & \textcolor{red}{12} & \textcolor{red}{15} & \textcolor{red}{18} & 18 & 18 & 18 & 18 \\ +4 & 4 & 4 & 4 & \textcolor{red}{4} & \textcolor{red}{8} & \textcolor{red}{12} & \textcolor{red}{16} & \textcolor{red}{20} & \textcolor{red}{24} & 24 & 24 & 24 & 24 \\ +5 & 5 & 5 & 5 & \textcolor{red}{5} & \textcolor{red}{10} & \textcolor{red}{15} & \textcolor{red}{20} & \textcolor{red}{25} & \textcolor{red}{30} & 30 & 30 & 30 & 30 \\ +6 & 6 & 6 & 6 & \textcolor{red}{6} & \textcolor{red}{12} & \textcolor{red}{18} & \textcolor{red}{24} & \textcolor{red}{30} & \textcolor{red}{36} & 36 & 36 & 36 & 36 \\ 6 & 6 & 6 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 36 & 36 & 36 \\ 6 & 6 & 6 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 36 & 36 & 36 \\ 6 & 6 & 6 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 36 & 36 & 36 \\ @@ -90,12 +91,12 @@ The command `padarray(A, Pad(:circular,4,4))` yields 12 & 16 & 20 & 24 & 4 & 8 & 12 & 16 & 20 & 24 & 4 & 8 & 12 & 16 \\ 15 & 20 & 25 & 30 & 5 & 10 & 15 & 20 & 25 & 30 & 5 & 10 & 15 & 20 \\ 18 & 24 & 30 & 36 & 6 & 12 & 18 & 24 & 30 & 36 & 6 & 12 & 18 & 24 \\ -3 & 4 & 5 & 6 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 1 & 2 & 3 & 4 \\ -6 & 8 & 10 & 12 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 2 & 4 & 6 & 8 \\ -9 & 12 & 15 & 18 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 3 & 6 & 9 & 12 \\ -12 & 16 & 20 & 24 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 4 & 8 & 12 & 16 \\ -15 & 20 & 25 & 30 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 5 & 10 & 15 & 20 \\ -18 & 24 & 30 & 36 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 6 & 12 & 18 & 24 \\ +3 & 4 & 5 & 6 & \textcolor{red}{1} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{6} & 1 & 2 & 3 & 4 \\ +6 & 8 & 10 & 12 & \textcolor{red}{2} & \textcolor{red}{4} & \textcolor{red}{6} & \textcolor{red}{8} & \textcolor{red}{10} & \textcolor{red}{12} & 2 & 4 & 6 & 8 \\ +9 & 12 & 15 & 18 & \textcolor{red}{3} & \textcolor{red}{6} & \textcolor{red}{9} & \textcolor{red}{12} & \textcolor{red}{15} & \textcolor{red}{18} & 3 & 6 & 9 & 12 \\ +12 & 16 & 20 & 24 & \textcolor{red}{4} & \textcolor{red}{8} & \textcolor{red}{12} & \textcolor{red}{16} & \textcolor{red}{20} & \textcolor{red}{24} & 4 & 8 & 12 & 16 \\ +15 & 20 & 25 & 30 & \textcolor{red}{5} & \textcolor{red}{10} & \textcolor{red}{15} & \textcolor{red}{20} & \textcolor{red}{25} & \textcolor{red}{30} & 5 & 10 & 15 & 20 \\ +18 & 24 & 30 & 36 & \textcolor{red}{6} & \textcolor{red}{12} & \textcolor{red}{18} & \textcolor{red}{24} & \textcolor{red}{30} & \textcolor{red}{36} & 6 & 12 & 18 & 24 \\ 3 & 4 & 5 & 6 & 1 & 2 & 3 & 4 & 5 & 6 & 1 & 2 & 3 & 4 \\ 6 & 8 & 10 & 12 & 2 & 4 & 6 & 8 & 10 & 12 & 2 & 4 & 6 & 8 \\ 9 & 12 & 15 & 18 & 3 & 6 & 9 & 12 & 15 & 18 & 3 & 6 & 9 & 12 \\ @@ -113,12 +114,12 @@ The command `padarray(A, Pad(:symmetric,4,4))` yields 12 & 9 & 6 & 3 & 3 & 6 & 9 & 12 & 15 & 18 & 18 & 15 & 12 & 9 \\ 8 & 6 & 4 & 2 & 2 & 4 & 6 & 8 & 10 & 12 & 12 & 10 & 8 & 6 \\ 4 & 3 & 2 & 1 & 1 & 2 & 3 & 4 & 5 & 6 & 6 & 5 & 4 & 3 \\ - 4 & 3 & 2 & 1 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 6 & 5 & 4 & 3 \\ - 8 & 6 & 4 & 2 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 12 & 10 & 8 & 6 \\ -12 & 9 & 6 & 3 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 18 & 15 & 12 & 9 \\ -16 & 12 & 8 & 4 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 24 & 20 & 16 & 12 \\ -20 & 15 & 10 & 5 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 30 & 25 & 20 & 15 \\ -24 & 18 & 12 & 6 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 36 & 30 & 24 & 18 \\ + 4 & 3 & 2 & 1 & \textcolor{red}{1} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{6} & 6 & 5 & 4 & 3 \\ + 8 & 6 & 4 & 2 & \textcolor{red}{2} & \textcolor{red}{4} & \textcolor{red}{6} & \textcolor{red}{8} & \textcolor{red}{10} & \textcolor{red}{12} & 12 & 10 & 8 & 6 \\ +12 & 9 & 6 & 3 & \textcolor{red}{3} & \textcolor{red}{6} & \textcolor{red}{9} & \textcolor{red}{12} & \textcolor{red}{15} & \textcolor{red}{18} & 18 & 15 & 12 & 9 \\ +16 & 12 & 8 & 4 & \textcolor{red}{4} & \textcolor{red}{8} & \textcolor{red}{12} & \textcolor{red}{16} & \textcolor{red}{20} & \textcolor{red}{24} & 24 & 20 & 16 & 12 \\ +20 & 15 & 10 & 5 & \textcolor{red}{5} & \textcolor{red}{10} & \textcolor{red}{15} & \textcolor{red}{20} & \textcolor{red}{25} & \textcolor{red}{30} & 30 & 25 & 20 & 15 \\ +24 & 18 & 12 & 6 & \textcolor{red}{6} & \textcolor{red}{12} & \textcolor{red}{18} & \textcolor{red}{24} & \textcolor{red}{30} & \textcolor{red}{36} & 36 & 30 & 24 & 18 \\ 24 & 18 & 12 & 6 & 6 & 12 & 18 & 24 & 30 & 36 & 36 & 30 & 24 & 18 \\ 20 & 15 & 10 & 5 & 5 & 10 & 15 & 20 & 25 & 30 & 30 & 25 & 20 & 15 \\ 16 & 12 & 8 & 4 & 4 & 8 & 12 & 16 & 20 & 24 & 24 & 20 & 16 & 12 \\ @@ -136,12 +137,12 @@ The command `padarray(A, Pad(:reflect,4,4))` yields 20 & 16 & 12 & 8 & 4 & 8 & 12 & 16 & 20 & 24 & 20 & 16 & 12 & 8 \\ 15 & 12 & 9 & 6 & 3 & 6 & 9 & 12 & 15 & 18 & 15 & 12 & 9 & 6 \\ 10 & 8 & 6 & 4 & 2 & 4 & 6 & 8 & 10 & 12 & 10 & 8 & 6 & 4 \\ -5 & 4 & 3 & 2 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 5 & 4 & 3 & 2 \\ -10 & 8 & 6 & 4 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 10 & 8 & 6 & 4 \\ -15 & 12 & 9 & 6 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 15 & 12 & 9 & 6 \\ -20 & 16 & 12 & 8 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 20 & 16 & 12 & 8 \\ -25 & 20 & 15 & 10 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 25 & 20 & 15 & 10 \\ -30 & 24 & 18 & 12 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 30 & 24 & 18 & 12 \\ +5 & 4 & 3 & 2 & \textcolor{red}{1} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{6} & 5 & 4 & 3 & 2 \\ +10 & 8 & 6 & 4 & \textcolor{red}{2} & \textcolor{red}{4} & \textcolor{red}{6} & \textcolor{red}{8} & \textcolor{red}{10} & \textcolor{red}{12} & 10 & 8 & 6 & 4 \\ +15 & 12 & 9 & 6 & \textcolor{red}{3} & \textcolor{red}{6} & \textcolor{red}{9} & \textcolor{red}{12} & \textcolor{red}{15} & \textcolor{red}{18} & 15 & 12 & 9 & 6 \\ +20 & 16 & 12 & 8 & \textcolor{red}{4} & \textcolor{red}{8} & \textcolor{red}{12} & \textcolor{red}{16} & \textcolor{red}{20} & \textcolor{red}{24} & 20 & 16 & 12 & 8 \\ +25 & 20 & 15 & 10 & \textcolor{red}{5} & \textcolor{red}{10} & \textcolor{red}{15} & \textcolor{red}{20} & \textcolor{red}{25} & \textcolor{red}{30} & 25 & 20 & 15 & 10 \\ +30 & 24 & 18 & 12 & \textcolor{red}{6} & \textcolor{red}{12} & \textcolor{red}{18} & \textcolor{red}{24} & \textcolor{red}{30} & \textcolor{red}{36} & 30 & 24 & 18 & 12 \\ 25 & 20 & 15 & 10 & 5 & 10 & 15 & 20 & 25 & 30 & 25 & 20 & 15 & 10 \\ 20 & 16 & 12 & 8 & 4 & 8 & 12 & 16 & 20 & 24 & 20 & 16 & 12 & 8 \\ 15 & 12 & 9 & 6 & 3 & 6 & 9 & 12 & 15 & 18 & 15 & 12 & 9 & 6 \\ @@ -161,12 +162,12 @@ The command `padarray(A, Fill(0,(4,4),(4,4)))` yields 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ -0 & 0 & 0 & 0 & \boxed{1} & \boxed{2} & \boxed{3} & \boxed{4} & \boxed{5} & \boxed{6} & 0 & 0 & 0 & 0 \\ -0 & 0 & 0 & 0 & \boxed{2} & \boxed{4} & \boxed{6} & \boxed{8} & \boxed{10} & \boxed{12} & 0 & 0 & 0 & 0 \\ -0 & 0 & 0 & 0 & \boxed{3} & \boxed{6} & \boxed{9} & \boxed{12} & \boxed{15} & \boxed{18} & 0 & 0 & 0 & 0 \\ -0 & 0 & 0 & 0 & \boxed{4} & \boxed{8} & \boxed{12} & \boxed{16} & \boxed{20} & \boxed{24} & 0 & 0 & 0 & 0 \\ -0 & 0 & 0 & 0 & \boxed{5} & \boxed{10} & \boxed{15} & \boxed{20} & \boxed{25} & \boxed{30} & 0 & 0 & 0 & 0 \\ -0 & 0 & 0 & 0 & \boxed{6} & \boxed{12} & \boxed{18} & \boxed{24} & \boxed{30} & \boxed{36} & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & \textcolor{red}{1} & \textcolor{red}{2} & \textcolor{red}{3} & \textcolor{red}{4} & \textcolor{red}{5} & \textcolor{red}{6} & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & \textcolor{red}{2} & \textcolor{red}{4} & \textcolor{red}{6} & \textcolor{red}{8} & \textcolor{red}{10} & \textcolor{red}{12} & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & \textcolor{red}{3} & \textcolor{red}{6} & \textcolor{red}{9} & \textcolor{red}{12} & \textcolor{red}{15} & \textcolor{red}{18} & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & \textcolor{red}{4} & \textcolor{red}{8} & \textcolor{red}{12} & \textcolor{red}{16} & \textcolor{red}{20} & \textcolor{red}{24} & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & \textcolor{red}{5} & \textcolor{red}{10} & \textcolor{red}{15} & \textcolor{red}{20} & \textcolor{red}{25} & \textcolor{red}{30} & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & \textcolor{red}{6} & \textcolor{red}{12} & \textcolor{red}{18} & \textcolor{red}{24} & \textcolor{red}{30} & \textcolor{red}{36} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ @@ -182,19 +183,21 @@ Each example is based on a multi-dimensional array ``\mathsf{A} \in\mathbb{R}^{2 ```math \mathsf{A}(:,:,1) = \boxed{ +\textcolor{red}{ \begin{array}{cc} 1 & 2 \\ 3 & 4 -\end{array}} +\end{array}}} \quad \text{and} \quad \mathsf{A}(:,:,2) = \boxed{ +\textcolor{red}{ \begin{array}{cc} 5 & 6 \\ 7 & 8 -\end{array}}. +\end{array}}}. ``` Note that each example will yield a new multi-dimensional array ``\mathsf{A}' @@ -220,16 +223,16 @@ The command `padarray(A, Pad(:replicate, 1, 1, 1))` gives: \boxed{ \begin{array}{cccc} 1 & 1 & 2 & 2 \\ -1 & \boxed{1} & \boxed{2} & 2 \\ -3 & \boxed{3} & \boxed{4} & 4 \\ +1 & \textcolor{red}{1} & \textcolor{red}{2} & 2 \\ +3 & \textcolor{red}{3} & \textcolor{red}{4} & 4 \\ 3 & 3 & 4 & 4 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 5 & 5 & 6 & 6 \\ -5 & \boxed{5} & \boxed{6} & 6 \\ -7 & \boxed{7} & \boxed{8} & 8 \\ +5 & \textcolor{red}{5} & \textcolor{red}{6} & 6 \\ +7 & \textcolor{red}{7} & \textcolor{red}{8} & 8 \\ 7 & 7 & 8 & 8 \end{array}} & @@ -262,16 +265,16 @@ The command `padarray(A, Pad(:circular, 1, 1, 1))` gives: \boxed{ \begin{array}{cccc} 4 & 3 & 4 & 3 \\ -2 & \boxed{1} & \boxed{2} & 1 \\ -4 & \boxed{3} & \boxed{4} & 3 \\ +2 & \textcolor{red}{1} & \textcolor{red}{2} & 1 \\ +4 & \textcolor{red}{3} & \textcolor{red}{4} & 3 \\ 2 & 1 & 2 & 1 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 8 & 7 & 8 & 7 \\ -6 & \boxed{5} & \boxed{6} & 5 \\ -8 & \boxed{7} & \boxed{8} & 7 \\ +6 & \textcolor{red}{5} & \textcolor{red}{6} & 5 \\ +8 & \textcolor{red}{7} & \textcolor{red}{8} & 7 \\ 6 & 5 & 6 & 5 \end{array}} & @@ -304,16 +307,16 @@ The command `padarray(A,Pad(:symmetric, 1, 1, 1))` gives: \boxed{ \begin{array}{cccc} 1 & 1 & 2 & 2 \\ -1 & \boxed{1} & \boxed{2} & 2 \\ -2 & \boxed{3} & \boxed{4} & 4 \\ +1 & \textcolor{red}{1} & \textcolor{red}{2} & 2 \\ +2 & \textcolor{red}{3} & \textcolor{red}{4} & 4 \\ 2 & 3 & 4 & 4 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 5 & 5 & 6 & 6 \\ -5 & \boxed{5} & \boxed{6} & 6 \\ -7 & \boxed{7} & \boxed{8} & 8 \\ +5 & \textcolor{red}{5} & \textcolor{red}{6} & 6 \\ +7 & \textcolor{red}{7} & \textcolor{red}{8} & 8 \\ 7 & 7 & 8 & 8 \end{array}} & @@ -346,16 +349,16 @@ The command `padarray(A, Pad(:reflect, 1, 1, 1))` gives: \boxed{ \begin{array}{cccc} 4 & 3 & 4 & 3 \\ -2 & \boxed{1} & \boxed{2} & 1 \\ -4 & \boxed{3} & \boxed{4} & 3 \\ +2 & \textcolor{red}{1} & \textcolor{red}{2} & 1 \\ +4 & \textcolor{red}{3} & \textcolor{red}{4} & 3 \\ 2 & 1 & 2 & 1 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 8 & 7 & 8 & 7 \\ -6 & \boxed{5} & \boxed{6} & 5 \\ -8 & \boxed{7} & \boxed{8} & 7 \\ +6 & \textcolor{red}{5} & \textcolor{red}{6} & 5 \\ +8 & \textcolor{red}{7} & \textcolor{red}{8} & 7 \\ 6 & 5 & 6 & 5 \end{array}} & @@ -390,16 +393,16 @@ The command `padarray(A, Fill(0, (1, 1, 1)))` gives: \boxed{ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ -0 & \boxed{1} & \boxed{2} & 0 \\ -0 & \boxed{3} & \boxed{4} & 0 \\ +0 & \textcolor{red}{1} & \textcolor{red}{2} & 0 \\ +0 & \textcolor{red}{3} & \textcolor{red}{4} & 0 \\ 0 & 0 & 0 & 0 \end{array}} \\ \mathsf{A}'(:,:,2) & = \boxed{ \begin{array}{cccc} 0 & 0 & 0 & 0 \\ -0 & \boxed{5} & \boxed{6} & 0 \\ -0 & \boxed{7} & \boxed{8} & 0 \\ +0 & \textcolor{red}{5} & \textcolor{red}{6} & 0 \\ +0 & \textcolor{red}{7} & \textcolor{red}{8} & 0 \\ 0 & 0 & 0 & 0 \end{array}} &