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Question:

If A = ⎛⎜⎝1020213⎞⎟⎠ and A³ - 6A² + 7A + kI₃ = 0, find k.

Solution:

A = ⎛⎜⎝1020213⎞⎟⎠, we get
A² = ⎛⎜⎝1020213⎞⎟⎠⎛⎜⎝1020213⎞⎟⎠ = ⎛⎜⎝5082458013⎞⎟⎠
A³ = ⎛⎜⎝5082458013⎞⎟⎠⎛⎜⎝1020213⎞⎟⎠ = ⎛⎜⎝210341282334055⎞⎟⎠
A³ - 6A² + 7A + kI₃ = 0, we get
⎛⎜⎝210341282334055⎞⎟⎠ - 6⎛⎜⎝5082458013⎞⎟⎠ + 7⎛⎜⎝1020213⎞⎟⎠ + k⎛⎜⎝100010001⎞⎟⎠ = 0
⎛⎜k𕒶000k𕒶000k𕒶⎞⎟⎠ = 0, which implies k = 0
Therefore the value of k is 2.