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Let A be a 3 × 3 matrix with eigenvalues 1, –1 and 3. Then
- a)A2 + 3A is non-singular
- b)A2 + A is singular
- c)A2 – A is non-singular
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Let A be a 3 × 3 matrix with eigenvalues 1, –1 and 3. Then...
A be a 3 × 3 matrix with eigenvalues of 1, –1 & 3.
For eigenvalues λ = 1, the characteristic equation is
|A – λI| = 0 ⇒ |A – I| = 0
⇒ |A2 – A|= 0 ⇒ A2 – A is singular
For λ = –1, the characteristic equation is |A + I| = 0 ⇒ |A2 + A| = 0 ⇒ A2 + A is singular
Similarly, for
λ = 3,
|A – 3I| = 0
⇒ |A2 – 3A| = 0 ⇒ A – 3A is singular
Since 0 & –3 are not eigenvalues,
So,|A| ≠ 0 & |A + 3I| ≠ 0
Hence |A2 + 3A| ≠ 0 ⇒ A2 + 3A is non-singular
For eigenvalues λ = 1, the characteristic equation is
|A – λI| = 0 ⇒ |A – I| = 0
⇒ |A2 – A|= 0 ⇒ A2 – A is singular
For λ = –1, the characteristic equation is |A + I| = 0 ⇒ |A2 + A| = 0 ⇒ A2 + A is singular
Similarly, for
λ = 3,
|A – 3I| = 0
⇒ |A2 – 3A| = 0 ⇒ A – 3A is singular
Since 0 & –3 are not eigenvalues,
So,|A| ≠ 0 & |A + 3I| ≠ 0
Hence |A2 + 3A| ≠ 0 ⇒ A2 + 3A is non-singular
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