How to show that a square matrix is not invertible if at least one row or column is zero ? I can show if a row is zero, the result C of $AB=C$ can not be the identity matrix because there is a zero row. But for the column case ?
Assume I don't know something about determinants.
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$\begingroup$Hint :
Let $A$ be a square matrix such that $i^{th}$ column is zero.
For any $B\in M_{n\times n}$ what would be the $i^{th}$ column of $BA$?
$\endgroup$ 4 $\begingroup$hint: theorem. let A be square invertible matrix. then [A,I] can be transformed into [I,A(inverse)] using elementary row operations.
but since A has a zero row or column, you can never transform the ith row or jth column be equal to 1 or that there will always be a zero in the diagonal which is not the identity matrix since [I]= 0 if i is not equal to j and 1 if i=j.
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