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19. True or false: If the n columns of S (eigenvectors of A) are independent, then (a) A is invertible. (b) A is diagonalizable. (c) S is invertible. (d) S is diagonalizable

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Martebi Martebi

Today at 8:24 AM

When n columns of S (eigenvectors of A) are independent, then:

(a) A is invertible is a false statement.

(b) A is diagonalizable is a true statement

Why is A not invertible?

Others are:

(c) S is invertible true statement.

(d) S is diagonalizable false statement.

Note that if n columns of S (eigen vectors of A) are said to be independent.

One can deduce or see that the eigen vector that is similar to the eigen value is said to be = 0 and one can say it is linearly independent when viewed with the other eigen vectors of A.

But because to the zero eigen value of A, we see that the determinant of A is the output of eigen values of A which is = 0 and as such, A is not invertible.

So, the linear independence of eigen vectors will not be able to find the invertibility of A and as such A is diagonalizable.

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