NIMCET 2020 — Mathematics PYQ

To determine which statement is not correct, let us evaluate each option based on the standard properties of determinants for square matrices of order $n$.

Checking Option A

• Property: For any diagonal matrix (or triangular matrix), the determinant is simply the product of its principal diagonal elements.

• Mathematical Statement: $\det (A)=a_{11}\cdot a_{22}\cdot \cdots \cdot a_{nn}$

• Status: This statement is correct.

Checking Option B

• Property: The determinant of the product of two square matrices of the same order equals the product of their individual determinants.

• Mathematical Statement: $\det (AB)=\det (A)\cdot \det (B)$

• Status: This statement is correct.

Checking Option C

• Property: When a square matrix $A$ of order $n$ is multiplied by a scalar $c$, every row (or column) of the matrix is multiplied by $c$. When evaluating its determinant, a factor of $c$ can be factored out from each of the $n$ rows.

• Mathematical Formula:

  $$\det (cA)=c^n\det (A)$$

• Given Statement: $\det (cA)=c[\det (A)]$

• Status: This statement is incorrect because it misses the power of $n$ (unless $n=1$).

Checking Option D

• Property: Transposing a matrix interchanges its rows and columns, which does not alter the value of its determinant.

• Mathematical Statement: $\det (A)=\det (A^T)$

• Status: This statement is correct.

Conclusion

The statement that is not correct is option C.

Correct Answer: C