Question

Suppose λ is a 3 x 3 matrix and ) is λ real number with the property that the equation Ax = λz is satisfied by some nonzero vector I. For each question, select true or false. If a statement does not even make sense, select false as your answer.

a. A - λI is not invertible.

Answer 1

The statement is true. A - λI is not invertible.

Step-by-step explanation:

To determine if A - λI is invertible, we need to check if it has an inverse.

If A - λI has an inverse, then (A - λI)^(-1) exists and (A - λI)(A - λI)^(-1) = I, where I is the identity matrix.

Let's assume that (A - λI) does have an inverse.

Then, (A - λI)(A - λI)^(-1) = I.

Multiplying these matrices, we have (A - λI)(A - λI)^(-1) = A(A - λI)^(-1) - λI(A - λI)^(-1) = I.

Simplifying this equation, we get A(A - λI)^(-1) - λI(A - λI)^(-1) = I.

Rearranging the terms, we have A(A - λI)^(-1) = λI(A - λI)^(-1) + I.

Since λ is a scalar, we can write it as a matrix: λI.

This equation can be further simplified to A(A - λI)^(-1)

= (λI + I)(A - λI)^(-1).

Now let's assume that (A - λI) does not have an inverse, which means that (A - λI)^(-1) does not exist.

If (A - λI)^(-1) does not exist, then A(A - λI)^(-1) does not exist either.

Therefore, A - λI is not invertible.