Can anyone check whether my following solution is correct and can anyone define the term "determinant"?

If det O =


|a b c


d e f


g h i|


=7





(a). det A =


| a b c


d e f


5g 5h 5j |


= 5 *


|a b c


d e f


g h i


= 5 * 7


= 35





(c). det C=


| a b c


g h i


d e f|


= -1*


| a b c


d e f


g h i |


= -1* 7


= -7





(d). det D =


| g h i


a b c


d e f|


= -1 *


| a b c


g h i


d e f|


= (-1) * (-1) *


|a b c


d e f


g h i|


= 7








(f). det F =


| a b c


3d 3e 3f


g h i|





= 3 *


| a b c


d e f


g h i|





= 3* 7





= 21








(e). det E =





| a b c


2d+a 2e+b 2f+c


g h i |





= det (E)^t


=


| a 2d+a g


b 2e+b h


c 2f+c i|


=


| a 2d g


b 2e h +


c 2f i|





| a a g


b b h


c c i|





= 2*


|a d g


b e h + 0 (since the first two columns are equal)


c f i|


= 2 *


| a b c


d e f


g h i|


= 2* 7


= 14

Can anyone check whether my following solution is correct and can anyone define the term "determinant"?
Yes, all the answers are correct.





A determinant is one of the scalar numbers associated with every square matrix. Each square matrix has a unique value of the determinant. Some concepts associated with a determinant:


If the determinant is nonzero, the matrix has an inverse.


If the determinant is nonzero, the denominator of its inverse is a factor of that determinant.





[Note: the determinant is not the only real number that can be associated with a matrix. some of these are the dimension of its vector field, the trace,...]


d:
Reply:tee hee hee
Reply:yes, all r correct