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Butoxors [25]
3 years ago
8

g Problem 2. (1.4 points) Suppose A is an n × n matrix, such that A 2 + 3A − 4I = 0 where I and 0 are the n × n identity and zer

o matrix respectively. What can you say about the invertibility of matrix A?
Mathematics
1 answer:
Pani-rosa [81]3 years ago
6 0

Answer:

The correct answer is A^{-1} = \frac{1}{4} A + \frac{3}{4 }<em> </em>I

Step-by-step explanation:

Since A is an n × n invertible non singular matrix, A^{-1} exists.

Given equation is A^{2} + 3A -4I = O where I and O are the n × n identity and zero matrix respectively.

⇒ A^{2} + 3A -4I = O

⇒ A^{-1} ( A^{2} + 3A -4I ) = A^{-1} × O

⇒ A^{-1} AA + 3 A^{-1}A - 4A^{-1}I = O

⇒ A + 3I - 4A^{-1} = O

⇒ A + 3I = 4 A^{-1}

⇒ A^{-1} = \frac{1}{4} A + \frac{3}{4 }<em> </em>I

Thus the value of  A^{-1} can be given by the above equation.

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