Please help me with the questions I got wrong.

Consider the 2 X 2 matrices They satisfy the properties that Do these relations look familiar? They sure do: they are exactly the same relations satised by the real numbers A 1 and 0 the complex numbers a 1 and l 0 the real numbers A land 1 the complex numbers i\" 0 and i. It follows that the 2 x 2 matrices contain the arithmetic of complex numbers! So for example the complex number 2 + 4 i would correspond to the matrix 3] E. 1 2 Note: the Maple notation for the matrix (3 4) is ,>. Consider the 2 x 2 matrices 1 I=( 0) and K=(0 1). 0 1 2 0 They satisfy the properties that .12: | :0 .K2= 2| co .IK: K :o .KI= K :0. Do these relations look familiar? They sure do: they are exactly the same relations satised by the complex numbers A 1 and 2i the real numbers A 1 and 0 0 the real numbers 9 1 and \\/ the complex numbers A 0 and i. It follows that the 2 X 2 matrices contain the arithmetic of certain kinds of irrational numbers! So for example the number 5 + 5v2 would correspond to the matrix 2 Note: the Maple notation for the matrix 3 4 is , >.Suppose that A = a b is a 2 X 2 matrix with the property that AX = XA for any 2 X 2 matrix X . We say that A commutes with any other matrix X . Looking at the special case X = 0 , we can deduce that b=0 as well as C=0 Looking at another special case X = ( ! ). , we can conclude (again) that C=0 O as well as a=d Putting these two facts together, we deduce that A must have a rather special form. Give a non-zero example of such a matrix A, whose elements are chosen from the set {0, 3, 7} A = X Note: the maple notation for the matrix 2 3 4 is , >.i) If A = -8 5 -4 - 6 then (AT) T = , > ii) If A and B are square matricies then (AB) is O equal to ATBT equal to BEAT . O undefined. iii) If C = (-5 1 ) then the matrix CCT = X while CP C is O the same as CCT O a vector a square matrix O undefined.For a general matrix A = a b c d we can see that A + A/ = is a symmetric matrix; while A - A/ = is a skew-symmetric + matrix. Now since A = -(A + A' ) + -(A - A! ), we can deduce that every square matrix A can be written as the sum of two symmetric matrices the sum of a symmetric and a skew-symmetric matrix the sum of two skew-symmetric matrices. Note: . the Maple notation for the matrix a b C d is , >. . include a space between the less than symbol