线性代数-初等矩阵，矩阵的秩

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✅ 原始矩阵 A：

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

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✳️ 初等变换矩阵左乘 A（行变换）

1. Eij：交换第 1 行 和 第 2 行

E =
[ 0  1  0 ]
[ 1  0  0 ]
[ 0  0  1 ]

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

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2. Ei(k)：第 2 行 × 2

E =
[ 1  0  0 ]
[ 0  2  0 ]
[ 0  0  1 ]

E × A =
[ 1   2   3 ]
[ 8  10  12 ]
[ 7   8   9 ]

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3. Eij(k)：第 3 行 ← 第 3 行 + 2 × 第 1 行

E =
[ 1  0  0 ]
[ 0  1  0 ]
[ 2  0  1 ]

E × A =
[ 1   2   3 ]
[ 4   5   6 ]
[ 9  12  15 ]

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✳️ A 右乘初等矩阵（列变换）

1. Eij：交换第 1 列 和 第 2 列

E =
[ 0  1  0 ]
[ 1  0  0 ]
[ 0  0  1 ]

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

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2. Ei(k)：第 3 列 × 3

E =
[ 1  0  0 ]
[ 0  1  0 ]
[ 0  0  3 ]

A × E =
[ 1   2   9  ]
[ 4   5  18 ]
[ 7   8  27 ]

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3. Eij(k)：第 2 列 ← 第 2 列 + 2 × 第 1 列

E =
[ 1  0  0 ]
[ 2  1  0 ]
[ 0  0  1 ]

A × E =
[ 5   2   3 ]
[14   5   6 ]
[23   8   9 ]

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继续

举例证明

这里面ij行互换的情况

看道例题

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