If a is matrix such that `A^(2)+A+2I=O`, then which of the following is/are true ?

Question

If a is matrix such that `A^(2)+A+2I=O`, then which of the following is/are true ?

If a is matrix such that `A^(2)+A+2I=O`, then which of the following is/are true ?
A. A is nonsingular
B. A is symmetric
C. A cannot be skew-symmetric
D. `A^(-1)=-1/2 (A+I)`

Monjur Alahi

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2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Salim Ahmed Kawsar

Given, `A^(2)+A+2I=O`
`implies A^(2)+A=-2I`
`implies |A^(2)+A|=|-2I|`
`implies |A||A+I|=(-2)^(n)`
`implies |A| ne 0`
Therefore, A is nonsingular, hence its inverse exists. Also, multiplying the given equation both sides with `A^(-1)`, we get
`A^(-1) =-1/2 (A+I)`

4 views Answered 2023-02-05 15:29