Question

Is T: M_2,2 → ℝ defined by T(A) =|A| a linear transformation? Provide a proof or counterexample.

Answer 1

i2 (0) Question: Let T:M32. > R be a function feom the set Maza to the Jef R, defined by T(A) = IAL -(1) whese M Mza is the set of all mateices of older axa and R is the set of que lead numbers. Also IAL the determinant of the mateix Solution: A function T:UF) → V(F) flow the vector space UIF) to the vector space VIF) over the same field F is said to be lineal teansformation if TQ+B) = T(Q)+ T(F) fa) and Tlax)=9T(*) 43) or combinedly Tlaq+6B) a T (Q)+bTIB) for every a, b E F and for every QBEU. = INS

(2) clearly the given function To Maja >R, defined by T(A) = 1 AL fol every AE Marzo As not a lineal transformation. For example we have A=To :)–sj (6) are the two different mateices in the i-eo, A = E Maja and B B= and B = 오의 set Maga [o Loi PR o A+B = :] = Ho Oto oto 0+1 A + B = **2=] - (7) !!

13) Now by definition of the function I given by (l), we have TIA) o = (1)0) – (0) (0),{by 07] AL T[ 1) = 0 -18) - Tỵ ) Agaj T(B) 131 O = 0) (1) – (0) (0) isky 1612 o o 1 is we fave T(B) (9 Again T(A+B) = 1 A+B), o; A,BE M2, .: A+B € M2, as Mas is a { by (7)} vector Space (1) (1)– (0) (0) =-1-0 = .: T (A+B) (10) Now feary equations (8) and (9), we

149 have T(4) FT (B) = 0+0 = 0 -o T(A)+. T(B) = 0 (11) Flow equations (10) and (II), we have T(A+B) = 1 I and T(A)+T(B) = 0 . We have T(4+B) + TTM) + T (B), (*1703 Since TIA+B) # TIA) + T(B), therefore the defined by T(A) = 1 AL for every A€ M2,2 is not a lineal transformation fimction T:M2, MR,