If A=\begin{bmatrix}1&2&2\\ 2&1&2\\ 2&2&1\end{bmatrix} then what is A^{2}-4A equal to?
- A. -5I_{3}
- B. -I_{3}
- C. I_{3}
- D. 5I_{3} ✓
Correct Answer: D. 5I_{3}
Explanation
Calculating A^2: the diagonal elements are 1(1)+2(2)+2(2) = 9, and off-diagonal elements are 1(2)+2(1)+2(2) = 8. For 4A, diagonal entries are 4 and off-diagonal entries are 8. Thus A^2 - 4A is a diagonal matrix with 9-4=5 on the main diagonal, so A^2 - 4A = 5I_3.
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