**"What is the matrix?"**

- Find a 2 by 2 matrix
**X**such that**X**^{2}= -**I**_{2}.*[1]* - Verify that if
**A**= a**I**_{2}+ b**X**for scalars a and b, then**A**is invertible and there are scalars d and c such that**A**^{-1}= d**I**_{2}+ c**X**.*[3]**[Note:***X**is a matrix satisfying the condition in problem 1 above.] - Find 4 by 4 matrices
**U**,**V**, and**W**such that**U**^{2}=**V**^{2}=**W**^{2}= -**I**_{4},**U****V**=**W**,**V****U**= -**W**,**V****W**=**U**,**W****V**= -**U**,**W****U**=**V**, and**U****W**= -**V**.*[2]* - Verify that if
**B**= a**I**_{2}+ b**U**+ c**V**+ d**W**for scalars a, b, c, and d, then**B**is invertible and there are scalars p, q, r, and s such that**B**^{-1}= p**I**_{2}+ q**U**+ r**V**+ s**W**.*[4]**[Note:***U**,**V**, and**W**are matrices satisfying the conditions in problem 3 above.]

Solutions to Assignment #4

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