[Math] Prove that if each row of a matrix sums to zero, then it has no inverse.

inverselinear algebramatrices

Could anyone help me with this proof without using determinant? I tried two ways.

Let $A$ be a matrix. If $A$ has the property that each row sums to zero, then there does not exist any matrix $X$ such that $AX=I$, where $I$ denotes the identity matrix.

I then get stuck. The other way was to prove by contradiction, and I failed too.

Best Answer

Hint: You can sum the elements of a row by multiplying this row with a vector of $1$'s. Can you find now a matrix $X$ (with appropriate columns) such that $AX=Ο$?

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