Consider the following statements: Statement-I: If X is an n n matrix, then (mX) = m^n (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 (Y) = m (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
Statement-I states that for an n n matrix X, (mX) = m^n (X). This is a standard property of determinants and is correct. Statement-II states that if a single row or column of a matrix is multiplied by a scalar m, the determinant of the resulting matrix is m times the determinant of the original matrix. This is also a fundamental property of determinants and is correct. Multiplying a matrix X by a scalar m is equivalent to multiplying each of its n rows by m. By applying the property given in Statement-II n times (once for each row), we can factor out m from each of the n rows, yielding m^n (X). Thus, Statement-II provides the logical reasoning required to prove Statement-I. Both statements are correct, and Statement-II explains Statement-I.
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