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

Step-by-step solution

Let (A= [ array ccc 1 & 0 & 0 0 & 1 & 0 0 & 0 & -1 array ] ).Then, (|A|=-1 ) If ( A ) is a prefect square of matrix (A₁ ). Then, (A₁^2=A ) ( |A₁ |^2=|A| |A₁ |^2=-1 A₁ ) cannot be a real matrix So, option correct. Let (R= [ array ccc -1 & 0 & 0 0 & -1 & 0 0 & 0 & -1 array ] ) be the sqare of a (3 3 ) matrix (B₁ ), Then, (B=B₁^2 |B₁ |^2=|B| |B₁ |^2= | array ccc -1 & 0 & 0 0 & -1 & 0 0 & 0 & -1 array |=-1 ) So, B cannot be the perfect square of a real matrix. In option, the given matrix is the identity matrix (I₃ ) su