Amari Flowers

2021-08-15

A is a 5x5 matrix. We know that there is a non-zero 5x1 vector B such that AB = 6B. Prove that A - 6I is not invertible

Mayme

Skilled2021-08-16Added 103 answers

It is given that A is a 5×5 matrix and there is a non-zero 5×1 vector B such that AB=6B that is (A−6I)B=0.

We have to prove that A−6I is not invertible. If possible let A−6I is invertible.

Then$(A-6I)}^{-1$ exist.

Now multiplying both side of$(A-6I)B=0by{(A-6I)}^{-1}we\ge t{(A-6I)}^{-1}(A-6I)B={(A-6I)}^{-1}0\Rightarrow B=0$

which contradicts that B is a non zero 5×1 matrix.

Thus A−6I is not invertible.

We have to prove that A−6I is not invertible. If possible let A−6I is invertible.

Then

Now multiplying both side of

which contradicts that B is a non zero 5×1 matrix.

Thus A−6I is not invertible.

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