Let A be a 5 times 5 matrix and suppose that detA4 For each

Let A be a 5 times 5 matrix and suppose that det(A)=4. For each of the following row operations, determine the value of det(B), where B is the matrix obtained by applying that row operation to A. Add 5 times row 3 to row 5 Interchange rows 1 and 4 Multiply row 3 by -4 Resulting values for det(B): det(5) = 0 det(5) = 0 det(5) = 0

Solution

we are applying certain operation on determinants

let determinant of A be det(A) which changes to det(B) after operation

so

first

a)   

add 5 times of row 3 to row 5

this operation doesnt change the value of determinant of A

so

det(E)=det(A)=4

b)interchange row 1 and 4

interchanging row lead to

     det(T)=-det(E)=-4

c) multiply the row 3 by -4

this will lead to

det(B)= -4*det(T)=-4*-4=16

so

det(B)=16 - answer

and given option are incorrect.