Which of the following statements is incorrect?

Option 4 is incorrect.

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Option 4 states: "If any two rows of a determinant are interchanged, then the sign of the determinant remains unchanged."

When two rows (or two columns) are swapped, the sign of the determinant changes to its opposite.

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Consider a 2×2 determinant:

$\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix}$

Determinant $= (1 \times 4) - (2 \times 3)$

$= 4 - 6$

$= -2$

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After swapping the two rows:

$\begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix}$

Determinant $= (3 \times 2) - (4 \times 1)$

$= 6 - 4$

$= +2$

The determinant changed from $-2$ to $+2$, showing the sign flips when rows are interchanged.

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Option 1 is correct: "Identical rows/columns → Determinant = 0"

$\begin{vmatrix} 2 & 3 \\ 2 & 3 \end{vmatrix} = (2 \times 3) - (3 \times 2) = 0$

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Option 2 is correct: "All zeros in one row → Determinant = 0"

$\begin{vmatrix} 0 & 0 \\ 5 & 7 \end{vmatrix} = (0 \times 7) - (0 \times 5) = 0$

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Option 3 is correct: "Rows ↔ Columns doesn't change value"

Original: $\begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = -2$

Transposed: $\begin{vmatrix} 1 & 3 \\ 2 & 4 \end{vmatrix} = -2$

Both determinants equal $-2$.

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Therefore, Option 4 is the incorrect statement.