```julia
using RowEchelon
using Latexify

rref([1 2 3; 4 5 6])

display(latexify(rref([1 2 3; 4 5 6])))

display(latexify(rref([1 2 3; 4 5 6])))
```


\begin{equation}
\left[
\begin{array}{ccc}
1.0 & 0.0 & -1.0 \\
0.0 & 1.0 & 2.0 \\
\end{array}
\right]
\end{equation}




\begin{equation}
\left[
\begin{array}{ccc}
1.0 & 0.0 & -1.0 \\
0.0 & 1.0 & 2.0 \\
\end{array}
\right]
\end{equation}




```julia
# The Four Subspaces for A
using RowEchelon
using Latexify

A = [1 3 5 0 7;
     0 0 0 1 2;
     1 3 5 1 9]

At = transpose(A)
display(A)
display(rref(A))

display("+++++++++++A Transpose+++++++++++")
display(At)
display(latexify(rref(At)))
```


    3×5 Matrix{Int64}:
     1  3  5  0  7
     0  0  0  1  2
     1  3  5  1  9



    3×5 Matrix{Float64}:
     1.0  3.0  5.0  0.0  7.0
     0.0  0.0  0.0  1.0  2.0
     0.0  0.0  0.0  0.0  0.0



    "+++++++++++A Transpose+++++++++++"



    5×3 transpose(::Matrix{Int64}) with eltype Int64:
     1  0  1
     3  0  3
     5  0  5
     0  1  1
     7  2  9



\begin{equation}
\left[
\begin{array}{ccc}
1.0 & 0.0 & 1.0 \\
0.0 & 1.0 & 1.0000000000000002 \\
0.0 & 0.0 & 0.0 \\
0.0 & 0.0 & 0.0 \\
0.0 & 0.0 & 0.0 \\
\end{array}
\right]
\end{equation}




```julia
A = [1 2 3;
     1 2 3;
     2 5 8]
display(A)
display(latexify(rref(A)))
```


    3×3 Matrix{Int64}:
     1  2  3
     1  2  3
     2  5  8



\begin{equation}
\left[
\begin{array}{ccc}
1.0 & 0.0 & -1.0 \\
0.0 & 1.0 & 2.0 \\
0.0 & 0.0 & 0.0 \\
\end{array}
\right]
\end{equation}




```julia
using RowEchelon
using Latexify

A = [1 2 3 1;
     1 1 2 1;
     1 2 3 1]
display(transpose(A))
transpose(A) * [-1,0,1]

```


    4×3 transpose(::Matrix{Int64}) with eltype Int64:
     1  1  1
     2  1  2
     3  2  3
     1  1  1





    4-element Vector{Int64}:
     0
     0
     0
     0




```julia
using RowEchelon
using Latexify

A = [1 3 4;  5 2 7]

rref(transpose(A))
```




    3×2 Matrix{Float64}:
     1.0  0.0
     0.0  1.0
     0.0  0.0




```julia
A = [1 1 1;
     1 0 1;
     1 -1 1]

display(rref(A))

A * [-1, 0, 1]
```


    3×3 Matrix{Float64}:
     1.0  0.0   1.0
     0.0  1.0  -0.0
     0.0  0.0   0.0





    3-element Vector{Int64}:
     0
     0
     0




```julia
A = [1 0 0;0 1 0;0 0 0]
A*A*A*A
```




    3×3 Matrix{Int64}:
     1  0  0
     0  1  0
     0  0  0




```julia
A = [1 0 0;0 1 0;0 0 0]
```




    3×3 Matrix{Int64}:
     1  0  0
     0  1  0
     0  0  0




```julia
a = [1.0, 2.0, 2.0]
b = [1.0, 1.0, 1.0]

x = 5/9

p = x * a

e = b - p

transpose(e) * a
```


    3-element Vector{Float64}:
      0.4444444444444444
     -0.11111111111111116
     -0.11111111111111116





    -2.220446049250313e-16




```julia

A = [1 2 2; 2 4 4; 2 4 4]
A = 1/9 * A
display(latexify(A))
display(latexify(A * A))
```


\begin{equation}
\left[
\begin{array}{ccc}
0.1111111111111111 & 0.2222222222222222 & 0.2222222222222222 \\
0.2222222222222222 & 0.4444444444444444 & 0.4444444444444444 \\
0.2222222222222222 & 0.4444444444444444 & 0.4444444444444444 \\
\end{array}
\right]
\end{equation}




\begin{equation}
\left[
\begin{array}{ccc}
0.1111111111111111 & 0.2222222222222222 & 0.2222222222222222 \\
0.2222222222222222 & 0.4444444444444444 & 0.4444444444444444 \\
0.2222222222222222 & 0.4444444444444444 & 0.4444444444444444 \\
\end{array}
\right]
\end{equation}




```julia
using RowEchelon
using Latexify

A = [1 2 3; 4 5 6; 7 8 9]

display(latexify(A))
display(latexify(A*A))
display(latexify(A*A*A))
display(latexify(A*A*A*A))
display(latexify(A*A*A*A*A))
```


\begin{equation}
\left[
\begin{array}{ccc}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{array}
\right]
\end{equation}




\begin{equation}
\left[
\begin{array}{ccc}
30 & 36 & 42 \\
66 & 81 & 96 \\
102 & 126 & 150 \\
\end{array}
\right]
\end{equation}




\begin{equation}
\left[
\begin{array}{ccc}
468 & 576 & 684 \\
1062 & 1305 & 1548 \\
1656 & 2034 & 2412 \\
\end{array}
\right]
\end{equation}




\begin{equation}
\left[
\begin{array}{ccc}
7560 & 9288 & 11016 \\
17118 & 21033 & 24948 \\
26676 & 32778 & 38880 \\
\end{array}
\right]
\end{equation}




\begin{equation}
\left[
\begin{array}{ccc}
121824 & 149688 & 177552 \\
275886 & 338985 & 402084 \\
429948 & 528282 & 626616 \\
\end{array}
\right]
\end{equation}




```julia
using LinearAlgebra

A = [0 1; 1 0]

eigvals(A)
```




    2-element Vector{Float64}:
     -1.0
      1.0




```julia
A = [.8 .3; .3 .7]
A*A*A*A*A*A*A*A*A*A
```




    2×2 Matrix{Float64}:
     0.986516  0.835441
     0.835441  0.708036




```julia

```

950be1817a2d3e3aeca5d707cd3ae356a9a44248
ipynb
Jupyter Notebook
julia/matrix/Matrix.ipynb
xiongxin/playground
9f23862b35c00b810e20f2e1e661142c024baa33
MIT
julia/matrix/Matrix.ipynb
xiongxin/playground
9f23862b35c00b810e20f2e1e661142c024baa33
MIT
julia/matrix/Matrix.ipynb
xiongxin/playground
9f23862b35c00b810e20f2e1e661142c024baa33
MIT
Qwen/Qwen-72B
1. YES
2. YES