Question

[ MATLAB] Check some linear algebra rules: Enter the following matrices:

Answer 1

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.