Step by step explanation along with Matlab code and output is provided below.

Step-by-step explanation:

We are given three matrices A, B, and C of size 2x2

A = [0 1; 0 0]

B =[1 2; -3 -6]

C =[4 -2; -2 1]

Output:

A = 0 1

0 0

B = 1 2

-3 -6

C = 4 -2

-2 1

Let us first check if the given matrices A, B, and C are singular or not

% the Matlab function det( ) calculates the determinant of a matrix

det_A=det(A)

det_B=det(B)

det_C=det(C)

Output:

det_A = 0

det_B = 3.3307e-16 (its practically zero)

det_C = 0

So the given matrices are singular which means that the determinant of the matrix is zero so inverse of these matrices is not possible.

Rule I:

a1=B*C

Output:

a1 = 0 0

0 0

In Matrix theory, if BC=0 then B=0 or C=0 doesn't hold true

For matrices B*C=0 does not imply that either B or C is zero matrix but rather it implies that at least one of them is singular. In this case we know that both B and C are singular matrices therefore, BC=0

Rule II:

a2=A^2

Output:

a2= 0 0

0 0

In Matrix theory, if A^2=0 then A=0 doesn't hold true

For matrices A^2=0 does not imply that A is zero matrix but rather it implies that A is singular. We already know that A is singular therefore, A^2=0

Rule III:

a3_L=(A+B)^2

a3_R=A^2+2*A*B+B^2

Output:

a3_L = -8 -15

15 27

a3_R = -11 -22

15 30

In Matrix theory, (A + B)^2 = A^2 + 2AB + B^2 doesn't hold true.

(A + B)^2 = A^2 + 2AB + B^2 might hold true if AB = BA, but generally, AB≠BA in matrix algebra.

Rule IV:

a4_L=(A-B)*(A+B)

a4_R=A^2-B^2

Output:

a4_L = 2 3

-15 -27

a4_R = 5 10

-15 -30

In Matrix theory, (A-B)(A+B) = A^2-B^2 doesn't hold true.

Rule V:

a5_L=A*(B+C)

a5_R=A*B+A*C

Output:

a5_L = -5 -5

0 0

a5_R = -5 -5

0 0

In Matrix theory, A(B+C) = AB+AC holds true.

Rule VI:

a6_L=A*(B+C)

a6_R=B*A+C*A

Output:

a6_L = -5 -5

0 0

a6_R = 0 5

0 -5

In Matrix theory, A(B+C) = BA+CA doesn't hold true.

on a side note; (B+C)A = BA+CA holds true

Rule VII:

a7_L=(A*B)^2

a7_R=A^2*B^2

Output:

a7_L = 9 18

0 0

a7_R = 0 0

0 0

In Matrix theory, (AB)^2 =A^2*B^2 doesn't hold true.

(AB)^2 =A^2*B^2 might hold true if and only if BA=AB which is not true in general.