Question

(1) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even.
(2) Compute the following determinant

(1) (4 pts) Prove or disprove that if all the elements of a matrix A is even, the determinant of A is even. (2) (2+2 pts) Compute the following determinant (123) (100 A= 1023 B=020 003 co c

Answer 1

let A be the mabix with all elements of A Solution: 0 are even . A= Call lari let aiz d22 913 923 1 --- Aik - ..dzic AR with all aij are even | a de acз Чек del (A) = All ri i 1 912 . | 922 923 --- Orie . 1921 92z - Qar) T L 1-9121 ? i ! | aka dk3 - .. dkk l lain ako ack/ +---+ Que 1 221 422 -- Uzey) 1 I din Orz arcon l Now determinant of each submahi x is an integer betty and 911, Che, . ax are even : Therefore even integer x integer = even integer and also addihon & substraction of even integer is also integer even integer. Hence here det CA) is ever

solution © let A=(1 2 37 1023 Loos det (m)= 1 31-216343%831 = 186 -070 det(A) = 6 let Balloon - lozol 1003 det (B) = 11331-0008/10/03/ det B) = 1XG det (B) = 67