Answer:
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.
