Question

if sigma is the eigenvalue of a nonsingular matrix A then what is the eigenvalue of A^-1​

Answer 1

Answer:

QUESTION:

If λλ is the eigen-value of a n×nn×n non-singular matrix AA and AA is a real orthogonal matrix, then prove that 1λ1λ is an eigen-value of the matrix AA.

MY ATTEMPT:

Since λλ is the eigen-value of a n×nn×n matrix AA, we have

|A−λIn|=0

|A−λIn|=0

Also since AA is a real orthogonal matrix,we have

AAT=ATA=In

AAT=ATA=In

So we can conclude that

|A−λ(AAT)|=0

|A−λ(AAT)|=0

Or,

|λA(1λIn−AT)|=0

|λA(1λIn−AT)|=0

Or,

|λA|⋅∣∣∣1λIn−AT∣∣∣=0

|λA|⋅|1λIn−AT|=0

Or, since AA is non-singular,

∣∣∣AT−1λIn∣∣∣=0

|AT−1λIn|=0

So, we can conclude that 1λ1λ is an eigen-value of the matrix ATAT.

But how do I prove that 1λ1λ is an eigen-value of the matrix AA?

Is my working faulty? Or is there a mistake in the question?

Please help.