Can we rely on Determinants for Linear Independence?

📌 CONCEPT: The linear independence of the column vectors of a matrix A is not solely determined by the value of its determinant d(A).

📐 RULE / FORMULA: The determinant d(A) can be zero even when the column vectors of A are linearly independent, and vice versa.

💡 WORKED EXAMPLE: Consider a matrix A with column vectors (1, 0, 0), (0, 1, 0), and (0, 0, 1). The determinant d(A) is 0, but the column vectors are linearly independent.

⚠️ COMMON MISTAKE: Many students incorrectly assume that a non-zero determinant implies linear independence of the column vectors, and vice versa.