For any 3×3 matrix M, let |M| denote the determinant of M. Let E=⎡⎢⎣12323481318⎤⎥⎦,P=⎡⎢⎣100001010⎤⎥⎦ and F=⎡⎢⎣13281813243⎤⎥⎦. If Q is a nonsingular matrix of order 3×3, then which of the following statements is(are) TRUE?
A
F=PEP and P2=⎡⎢⎣100010001⎤⎥⎦
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B
|EQ+PFQ−1|=|EQ|+|PFQ−1|
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C
|(EF)3|>|EF|2
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D
Sum of the diagonal entries of P−1EP+F is equal to the sum of diaginal entries of E+P−1FP
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Solution
The correct option is D Sum of the diagonal entries of P−1EP+F is equal to the sum of diaginal entries of E+P−1FP Clearly, P2=⎡⎢⎣100010001⎤⎥⎦=1 and PE=⎡⎢⎣12381318234⎤⎥⎦ ⇒PEP=⎡⎢⎣13281813243⎤⎥⎦=F