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Chapter 7: Q7-30E (page 310)

30. If two n x n matrices A and B are diagonalizable, then A+B must be diagonalizable as well.

Short Answer

Expert verified

The statement is False

Step by step solution

01

Define diagonalizable matrix

If any triangular matrix of order n has n distinct eigenvalues, then matrix A is diagonalizable.

02

Explanation:

For example, consider two upper triangular matrices.

A=-1100,B=1000

Clearly, both matrices are diagonalizable because both have distinct eigenvalues.

Now, find the sum of both the matrices.

A+B=0100

Which is not diagonalizable, because it has only one eigenvalue, that is 0.

So, the given statement is 0.

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Most popular questions from this chapter

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

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