null matrix and invertible matrix

seema283k

how can we use property of null matrix .explain it by giving examples.
please give me explanation of the theorem----if A is a null matrix & B is any matrix then I+AB is invertible iff I+BA is invertible

ygolo

Are you sure you mean the null matrix?

Because the null matrix is the matrix with all zero entries.

So if A is the null matrix, then I+AB=I=I+BA. I is invertible. So for your theorem, both the hypothesis and conclusion are true (in both directions). Making the theorem vacuously true.

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seema283k	null matrix and invertible matrix how can we use property of null matrix .explain it by giving examples. please give me explanation of the theorem----if A is a null matrix & B is any matrix then I+AB is invertible iff I+BA is invertible

ygolo	Are you sure you mean the null matrix? Because the null matrix is the matrix with all zero entries. So if A is the null matrix, then I+AB=I=I+BA. I is invertible. So for your theorem, both the hypothesis and conclusion are true (in both directions). Making the theorem vacuously true.