{
  "A": [
    [
      0,
      2,
      1
    ],
    [
      1,
      1,
      1
    ],
    [
      2,
      3,
      4
    ]
  ],
  "b": [
    7,
    6,
    20
  ],
  "L": [
    [
      "1",
      "0",
      "0"
    ],
    [
      "0",
      "1",
      "0"
    ],
    [
      "1/2",
      "-1/4",
      "1"
    ]
  ],
  "U": [
    [
      "2",
      "3",
      "4"
    ],
    [
      "0",
      "2",
      "1"
    ],
    [
      "0",
      "0",
      "-3/4"
    ]
  ],
  "row_permutation_PA_order": [
    2,
    0,
    1
  ],
  "inverse_permutation_for_A_equals_PLU": [
    1,
    2,
    0
  ],
  "pivot_trace": [
    {
      "column": 0,
      "pivot_row_before_swap": 2,
      "row_order_after": [
        2,
        1,
        0
      ],
      "swapped_previous_L_columns": [],
      "eliminated": [
        {
          "row": 1,
          "multiplier": "1/2"
        },
        {
          "row": 2,
          "multiplier": "0"
        }
      ]
    },
    {
      "column": 1,
      "pivot_row_before_swap": 2,
      "row_order_after": [
        2,
        0,
        1
      ],
      "swapped_previous_L_columns": [
        0
      ],
      "eliminated": [
        {
          "row": 2,
          "multiplier": "-1/4"
        }
      ]
    },
    {
      "column": 2,
      "pivot_row_before_swap": 2,
      "row_order_after": [
        2,
        0,
        1
      ],
      "swapped_previous_L_columns": [],
      "eliminated": []
    }
  ],
  "Pb": [
    "20",
    "7",
    "6"
  ],
  "forward_y": [
    "20",
    "7",
    "-9/4"
  ],
  "backward_x": [
    "1",
    "2",
    "3"
  ],
  "independent_Leibniz_det": "-3",
  "swap_count": 2,
  "upper_diagonal_product": "-3",
  "determinant_permutation_sign_checked": true,
  "independent_Cramer_solution": [
    "1",
    "2",
    "3"
  ],
  "wrong_without_previous_L_swap": {
    "L": [
      [
        "1",
        "0",
        "0"
      ],
      [
        "1/2",
        "1",
        "0"
      ],
      [
        "0",
        "-1/4",
        "1"
      ]
    ],
    "U": [
      [
        "2",
        "3",
        "4"
      ],
      [
        "0",
        "2",
        "1"
      ],
      [
        "0",
        "0",
        "-3/4"
      ]
    ],
    "PA_equals_LU": false
  },
  "invalid_inputs_rejected": [
    "singular",
    "ragged",
    "empty",
    "nonfinite",
    "bool"
  ],
  "scope": "own exact PA=LU; SciPy uses inverse permutation A=P L U; no SciPy execution/floating stability/sparse solver/benchmark"
}