Question

Let A be a 2 × 2 matrix with real entries. Let I be the 2 × 2 identity matrix. Denote by tr (A), the sum of diagonal entries of A. Assume that A₂= I.
Statement −1: If A ≠ I and A ≠ − I, then det A = − 1.
Statement −2: If A ≠ I and A ≠ − I, then tr (A) ≠ 0.

Statement −1 is false, Statement −2 is true

Statement −1 is true, Statement −2 is true, Statement −2 is a correct explanation for Statement −1

Statement −1 is true, Statement −2 is true; Statement −2 is not a correct explanation for Statement −1.

Statement − 1 is true, Statement − 2 is false.

Solution

Statement − 1 is true, Statement − 2 is false.

Let space straight A space equals space open square brackets table row straight a straight b row straight c straight d end table close square brackets space so space that space straight A squared space equals space open square brackets table row cell straight a squared plus bc end cell cell ab space plus space bd end cell row cell ac space plus dc end cell cell bc plus straight d squared end cell end table close square brackets space equals space open square brackets table row 1 0 row 0 1 end table close square brackets rightwards double arrow space straight a squared space plus bc space equals space 1 space equals space bc plus straight d squared and space left parenthesis straight a plus straight d right parenthesis straight c space equals space 0 space left parenthesis straight a plus straight d right parenthesis straight b. Since space space straight A space not equal to space straight I comma space straight A space not equal to space 1 comma space straight a space equals space minus straight d space and space hence space det space straight A space equals space open vertical bar table row cell square root of 1 minus bc end root end cell straight b row straight c cell negative square root of 1 minus bc end root end cell end table close vertical bar space equals space minus 1 plus bc minus bc space equals space minus 1 Statement space 1 space is space true But space straight A space equals space 0 space space and space hence space statement space 2 space is space false.

For Daily Free Study Material Join wiredfaculty