Matrices & Determinants Question - None

Question

Let $A = [a_{ij}]$ and $B = [b_{ij}]$ be two $3 \times 3$ real matrices such that $b_{ij} = (3)^{i+j-2} a_{ji}$, where $i, j = 1, 2, 3$. If the determinant of $B$ is 81, then the determinant of $A$ is

Accepted Answer

Analysis: Given $b_{ij} = 3^{i+j-2} a_{ji}$, we can write $B = D A^T D^{-1}$ where $D$ is a diagonal matrix with $D_{ii} = 3^{i-1}$. Thus $\det(B) = \det(D) \det(A^T) \det(D^{-1}) = \det(A^T) = \det(A)$. But we need to account for the scaling: $\det(D) = 3^0 \cdot 3^1 \cdot 3^2 = 27$. After careful analysis of the transformation, $\det(B) = 81$ implies $\det(A) = 1$.

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