InterviewSolution
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InterviewSolution
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Which of the following is (are) NOT the square of a `3xx3`matrix with real entries?`[1 0 0 0 1 0 0 0-1]`(b) `[-1 0 0 0-1 0 0 0-1]``[1 0 0 0 1 0 0 0 1]`(d) `[1 0 0 0-1 0 0 0-1]`A. `{:[(1,0,0),(0,1,0),(0,0,-1)]:}`B. `{:[(-1,0,0),(0,-1,0),(0,0,-1)]:}`C. `{:[(1,0,0),(0,1,0),(0,0,1)]:}`D. `{:[(-1,0,0),(0,-1,0),(0,0,-1)]:}` |
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Answer» Correct Answer - A::B Let `{:A=[(1,0,0),(0,1,0),(0,0,-1)]:}`.Then, `absA=-1`. If A is a prefect square of matrix `A_1`. Then, `A_(1)^2=A` `rArr abs(A_1)^2=absArArr abs(A_1)^2=-1rArr A_1` cannot be a real matrix So, option(a) correct. Let `{:R=[(-1,0,0),(0,-1,0),(0,0,-1)]:}` be the sqare of a `3xx3` matrix `B_1`, Then, `B=B_1^2rArrabs(B_1)^2=absBrArrabs(B_1)^(2) = {:abs((-1,0,0),(0,-1,0),(0,0,-1))=-1:}` So, B cannot be the perfect square of a real matrix. In option (c), the given matrix is the identity matrix `I_3` such that `I_3=I_3^2` Conisder the matrix `{:[(1,0,0),(0,-1,0),(0,0,-1)]:}` given in option (d). Clearly, `{:[(1,0,0),(0,-1,0),(0,0,-1)]=[(1,0,0),(0,0,1),(0,-1,0)][(1,0,0),(0,0,1),(0,-1,0)]:}` Thus, the matrix given in option(d) is the perfect square of a real matrix. |
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