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Say it is a 3x3 matrix. Can you multiply it with a 3x3 determinant without expanding and calculating the determinant's value? Thank you.

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In general, no, unless if the matrix is a zero matrix.

Simply because if there exists an algorithm that can, for a pair of matrices $A\neq 0, B$, calculate the matrix $C=\det(B)\cdot A$, then that algorithm can be used to calculate the determinant of $B$. That's because there exists a pair of indices $i,j$ such that $A_{i,j}\neq0$, meaning that $$C_{i,j}=\det(B)\cdot A_{i,j}$$

and $$\det(B)=\frac{C_{i,j}}{A_{i,j}}$$

This means that any algorithm that can calculate $\det(B)\cdot A$ can also calculate $\det(B)$.

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