Question

Let \(A=[a_{ij}]\), \(\det(A)\neq 0\), and \(B=[b_{ij}]\) be two \(3\times 3\) matrices. If \[ b_{ij}=3^{\,i-j}\,a_{ij}\quad \text{for all } i,j=1,2,3, \] then:

• A. \(3\det(A)=\det(B)\)
• B. \(27\det(A)=\det(B)\)
• C. \(\det(A)=\det(B)\)
• D. \(\det(A)=27\det(B)\)

Hint:

When each element is multiplied by \(k^{i-j}\), separate the effect into \textbf{row scaling} and \textbf{column scaling}. If total row and column powers cancel, the determinant remains unchanged.

Answer 1

The Correct Option is C