The matrix 4A has an eigenvalue Solutionv is eigen vector of

The matrix ?4A has an eigenvalue _________

Solution

v is eigen vector of A

1) A^3 v = A^2 (Av)
A^3 v= A^2 (4v)
A^3 v= 4 A^2 Av
. A^3 v.= 4 A (4v)
A^3 v= 4^2 A v
.
A^3 v = 4^3 v.

So, 4^3 is an eigenvalue of A^3 that is 64
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2) Av = Lv => A^-1 A v = A^-1 Lv = > A^-1 v = 1/L v => A^-1 has eigne value of 1/L = 1/4 .

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3) (A + 4I)v = Av + 4Iv = 4v + 4v = 8v.
So, 8 is an eigenvalue of A + 4I.
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4) -4Av = -4(8v) = -32v.
So, -32 is an eigenvalue of -4A.
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