Consider the following statements: Statement-I: If X is an n \times n matrix, then \det(mX)=m^n \det(X), where m is a scalar. Statement-II: If Y is a matrix obtained from X by multiplying any row or column by a scalar m, then \det(Y)=m \det(X). Which one of the following is correct in respect of the above statements?
- A. Both Statement-I and Statement-II are correct and Statement-II explains Statement-I ✓
- B. Both Statement-I and Statement-II are correct but Statement-II does not explain Statement-I
- C. Statement-I is correct but Statement-II is not correct
- D. Statement-I is not correct but Statement-II is correct
Correct Answer: A. Both Statement-I and Statement-II are correct and Statement-II explains Statement-I
Explanation
Statement-II provides a fundamental property of determinants (factoring out a scalar from a single row/column). Statement-I is a direct consequence of applying Statement-II n times (once for each of the n rows) when the entire matrix is scaled by m. Thus, both are correct, and II explains I.
Related questions on Matrices & Determinants
- Consider the determinant \Delta=\begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{vmatrix} If $a_{13...
- If A=\begin{pmatrix}1&0&0\\ 0&\cos~\theta&\sin~\theta\\ 0&\sin~\theta&-\cos\theta\end{pmatrix}, then which of the following are correct?...
- If X is a matrix of order 3\times3, Y is a matrix of order 2\times3 and Z is a matrix of order 3\times2, then which of the follo...
- What is the value of a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}?
- What is the value of a_{21}C_{11}+a_{22}C_{12}+a_{23}C_{13}?