For the matrix X shown in the image, if X2 − X + I = O, where I is the…
2004
For the matrix X shown in the image, if X2 − X + I = O, where I is the identity matrix and O is the zero matrix, then X−1 is:
Answer: B. B — CONCEPTIf a square matrix X satisfies a polynomial identity that can be rearranged into XQ(X) = I, then Q(X) is an inverse of X. An inverse is unique, so…
- A.
A
- B.
B
- C.
C
- D.
D
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Correct answer: B
CONCEPT
If a square matrix X satisfies a polynomial identity that can be rearranged into XQ(X) = I, then Q(X) is an inverse of X. An inverse is unique, so proving either XQ = I or QX = I determines X−1.
Because every polynomial in X commutes with X, the rearrangement can be performed without changing factor order.
APPLICATION
Start from X2 − X + I = O.
Move the identity term: X2 − X = −I.
Factor X: X(X − I) = −I.
Multiply by −1: X(I − X) = I.
Therefore X−1 = I − X.
For X = [[a, 1], [−a2+a−1, 1−a]], subtract each entry from the corresponding entry of I to obtain [[1−a, −1], [a2−a+1, a]].
CROSS-CHECK
Multiplying X by [[1−a, −1], [a2−a+1, a]] gives [[1, 0], [0, 1]] after cancellation. Hence the inverse is [[1−a, −1], [a2−a+1, a]].