Which of the following is (are) NOT the square of a 3 × 3 matrix with real entries?

Which of the following is (are) NOT the square of a 3×3 matrix with real entries?
  1. 1000-1000-1
  2. 10001000-1
  3. 100010001
  4. -1000-1000-1

Solution

Let \(A=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1\end{array}\right]\).Then, \(|A|=-1\) If \(\mathrm{A}\) is a prefect square of matrix \(A_1\). Then, \(A_1^2=A\) \(\Rightarrow\left|A_1\right|^2=|A| \Rightarrow\left|A_1\right|^2=-1 \Rightarrow A_1\) cannot be a real matrix So, option correct. Let \(R=\left[\begin{array}{ccc}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1\end{array}\right]\) be the sqare of a \(3 \times 3\) matrix \(B_1\), Then, \(B=B_1^2 \Rightarrow\left|B_1\right|^2=|B| \Rightarrow\left|B_1\right|^2=\left|\begin{array}{ccc} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right|=-1\) So, B cannot be the perfect square of a real matrix. In option, the given matrix is the identity matrix \(I_3\) such that \(I_3=I_3^2\) \(\begin{aligned} &\text { Conisder the matrix }\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right] \text { given in option. Clearly, } \\ &{\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array}\right]=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{array}\right]\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{array}\right]} \end{aligned}\) Thus, the matrix given in option is the perfect square of a real matrix.

Asked in: JEE Advanced 2017 (Paper 1)

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