{
  "schema": "ml-prep/item@1",
  "item": {
    "id": "original-la-foundations",
    "area": "mathematics",
    "topic": "linear-algebra",
    "origin": "original",
    "title": "Matrix fundamentals under time pressure",
    "skills": [
      "matrix-operations",
      "inverse",
      "determinant-trace",
      "matrix-norms"
    ],
    "priority": "foundation",
    "difficulty": "easy",
    "estimated_minutes": 8,
    "prerequisites": [
      "matrix-arithmetic"
    ],
    "prompt": "Let\n$$\nA = \\begin{bmatrix} 1 & 2 \\\\ 3 & 4 \\end{bmatrix}.\n$$\nCompute $\\det(A)$, $\\operatorname{tr}(A)$, $A^{-1}$, and the Frobenius norm $\\lVert A \\rVert_F$. State whether $A$ is invertible and justify the answer in one sentence.\n",
    "answer": "$$\n\\det(A) = -2, \\qquad \\operatorname{tr}(A) = 5, \\qquad\nA^{-1} = \\begin{bmatrix} -2 & 1 \\\\ \\frac{3}{2} & -\\frac{1}{2} \\end{bmatrix}, \\qquad\n\\lVert A \\rVert_F = \\sqrt{30}.\n$$\n$A$ is invertible because its determinant is nonzero.\n",
    "check": {
      "kind": "numeric",
      "id": "original-la-foundations",
      "values": [
        -2,
        5,
        -2,
        1,
        1.5,
        -0.5,
        30
      ]
    }
  }
}