Question

Let T:R2-R2 be multiplication by A. Determine whether T has an inverse; if so, find X2 5 81 -3 3 A- If inverse exists enter y1 and y2, otherwise enter NA for both. Click here to enter or edit your answer

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Answer 1

We know that {e₁,e₂} = {(1,0)^T,(0,1)^T} is the standard basis for R². Further, T(e₁) = A(1,0)^T = (5,-3)^T and T(e₂) = A(0,1)^T = (8,3)^T. Hence, the standard matrix of T is M = [T(e₁),T(e₂)] =

5	8
-3	3

Further, since det(M)= 15+24 = 39 ? 0, hence M is invertible. Therefore, T^-1 exists and T^-1: R²? R² is defined by T^-1 (x₁,x₂)^T = M^-1 . (x₁,x₂)^T.

Now, M^-1 =

1/13	-8/39
1/13	5/39

so that T^-1(x₁,x₂)^T=M^-1.(x₁,x₂)^T=(x₁/13-8x₂/39, x₁/13+5x₂/39)^T. Thus, y₁= x₁/13-8x₂/39 and y₂= x₁/13+5x₂/39.

The Chegg software is deleting all the mathematical symbols and replacing these by ?. Hence a screenshot is also attached.