Verified answer

LET X be any 3*3 matix

X = [abc/def/hij]

O = [000/000/000]

X+O = [abc/def/hij] + [000/000/000] =

[a+0 b+0 c + 0 / d+0 e+0 f+0/ h+0 i+0 j+0] = X

&

O + X = [000/000/000] + [abc/def/hij] =

[0+a 0+b 0+c/ 0+d 0+e 0+f/ 0+h 0+ i 0+j] = X

so

X + O = X

O + X = X

thus

O is a an additive Identity

this is an outline

you need to qoute your definition of an additive Identity

It's kind of intuitive. But I guess to prove it, you would have to show that an arbitrary 3*3 matrix added to the zero matrix leaves the original 3*3 matrix unchanged. That is the definition of the identity. (i.e. a + e = a = e + a) So first pick an arbitrary 3*3 matrix. Say,

[ a b c

d e f

g h i ]

where a, b, c, d, e, f, g, h, i are integers. Then,

[ a b c

d e f

g h i ]

+

[ 0 0 0

0 0 0

0 0 0 ]

=

[ a+0 b+0 c+0

d+0 e+0 f+0

g+0 h+0 i+0 ]

=

[ a b c

d e f

h i j ]

Since adding the zero matrix to any arbitrary 3*3 matrix leaves the matrix unchanged, it is the identity matrix. You can also show that order doesn't matter.

Matrix addition is commutative and associative. Matrix multiplication is neither. a nil matrix is a matrix with all entries as 0. An identity matrix is a sq. matrix with the important diagonal all 1s and something of the entries 0s.