If A={1 2} B={2 1} C {-3 1}
{-2 3 } { 2 3} {2 3}
Verify that ABC=A(BC)
And A(B+C)=AB+AC
To verify that ABC = A(BC) and A(B+C) = AB+AC, we must first perform the necessary matrix multiplications.
1. ABC = A(BC):
First, we will find BC by multiplying matrices B and C.
BC = {{2, 1}, {2, 3}} * {{-3, 1}, {2, 3}}
= {{(2*-3) + (1*2), (2*1) + (1*3)}, {(2*-3) + (3*2), (2*1) + (3*3)}}
= {{-6+2, 2+1}, {-6+6, 2+9}}
= {{-4, 3}, {0, 11}}
Next, we will multiply matrix A with BC.
A(BC) = {{1, 2}} * {{-4, 3}, {0, 11}}
= {{(1*-4) + (2*0), (1*3) + (2*11)}}
= {{-4, 25}}
Comparing ABC and A(BC), we can see that {{-4, 25}} is equal to {{-4, 25}}, so ABC = A(BC).
2. A(B+C) = AB + AC:
First, we will find the sum of B and C.
B + C = {{2, 1}, {2, 3}} + {{-3, 1}, {2, 3}}
= {{2+(-3), 1+1}, {2+2, 3+3}}
= {{-1, 2}, {4, 6}}
Next, we will multiply matrix A with B and C separately, and then find their sum.
AB = {{1, 2}} * {{2, 1}, {2, 3}}
= {{(1*2) + (2*2), (1*1) + (2*3)}}
= {{4, 7}}
AC = {{1, 2}} * {{-3, 1}, {2, 3}}
= {{(1*-3) + (2*2), (1*1) + (2*3)}}
= {{1, 7}}
Adding AB and AC, we get:
AB + AC = {{4, 7}} + {{1, 7}}
= {{4+1, 7+7}}
= {{5, 14}}
Comparing A(B+C) and AB + AC, we can see that {{5, 14}} is equal to {{5, 14}}, so A(B+C) = AB + AC.
Therefore, we have verified that ABC = A(BC) and A(B+C) = AB+AC.