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Chapter 7: Q44E (page 383)

If A is a diagonalizable 4 脳 4 matrix with A4=0, then A must be the zero matrix.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Definition of diagonalizable

Any matrix A is said to be diagonalizable, if and only if P-1AP is diagonal.

02

Explanation for A must be the zero

Consider a 4 脳 4 matrix A is diagonalizable, then there exists an invertible matrix S and a diagonal matrix D such that S-1AS = D.

From this we have D4 = S-1A4S = S-10S = 0, Since D is a diagonal matrix, we must have must D=0 and therefore A = SDS-1= 0.

So, the given statement is true.

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