TITLE: Can we use simultaneous row and column operations in solving same determinant?
QUESTION [6 upvotes]: Please help.. Can both(row and column) operations be used simultaneously in finding the value of same determinant means in solving same question at a single time?

REPLY [4 votes]: A row operation corresponds to multiplying a matrix $A$ on the left by one of several elementary matrices whose determinants are easy to compute to get a matrix $B = EA$. For instance, swapping the rows of a 2x2 matrix is done with 
$$
\pmatrix{0 & 1 \\ 1 & 0 } \pmatrix{a & b \\ c & d}
$$
The determinant of the resulting row-swapped matrix is the product of the two determinants. Hence $det B = det E det A$. Since $det E = -1$ in this case, you can compute the det of the new matrix and multiply it be $-1$ to get the det of the original one. (This is typically not very useful, but it's an example). 
In the same way, a column op is done with $A \mapsto AE$, and you can use the same rule -- prduct of determinants -- to relate the determinant of $B = AE$ to the determinant of $A$. 
In short: you can do a sequence of row and column ops, each of which adds a factor to the determinant, until you reach the identity. You don't have to do just a sequence of row ops or just a sequence of column ops. 
Personal advice: Just use one or the other. It'll take a little longer, but you're much less likely to make a mistake in my experience with many students over the years.

REPLY [3 votes]: Remember that a set of row operations on a matrix $A$ can be represented by a matrix $P$
$$A^{\prime}=PA$$
Here $A^{\prime}$ is the matrix $A$ after the operations. In the same way, column operations are represented by a matrix $Q$ such that
$$A^{\prime}=AQ$$
After both row a column operations you have
$$A^{\prime}=PAQ$$
so
$$\det A^{\prime}=\det P\det Q\det A$$
Here $\det P$ and $\det Q$ are factors that depends on your exact operations: row addition (factor $1$), row transposition (factor $-1$) and row multiplication by $c$ (factor $c^{n}$, where $n\times n$ is the size of the matrices).