[Math] Let $V$ be the vector space of all $4$x$4$ matrices such that the sum of the elements in any row or any column is the same.

linear algebramatricesproof-verification

Let $V$ be the vector space of all $4$x$4$ matrices such that the sum of the elements in any row or any column is the same. What is the dimension of $V$?

Sol: I thought of this matrix where every row and column sums to $s$ and since it has $10$ variables I think the dim is 10. By separating and taking out the variables I could come up with a $10$ element basis. Through an obvious but lengthy process I could show its linear independence and the fact that it's a spanning set is obvious from the construction. Is this correct?
\begin{bmatrix}
a & b & c & s-(a+b+c)\\
d & e & f & s-(d+e+f)\\
g & h & i & s-(g+h+i)\\
s-(a+d+g) & s-(b+e+h) & s-(c+f+i) &-2s+(a+b+c+d+e+f+g+h+i)
\end{bmatrix}

Best Answer

In general, the dimension of the subspace of $M_n(K)$ with equal row and column sum is $(n-1)^2+1$, see here:

Dimension of vector space of matrices with zero row and column sum.

Actually, if the value is supposed to be zero, the dimension is $(n-1)^2$. This follows for $m=n$ from the duplicate. We have to add plus $1$, if the value is not prescribed.