Can someone check my answer?

Given matrices A,B, and C are all 2 x 2, determine whether the equation is true for the given matrices.
C(A+B) = AC +BC
My answer: False

Matrix multiplication is not commutative.
My answer: true

For matrices, A + B = B + A
My answer: true

I agree.

Thanks

To check whether the equation C(A+B) = AC + BC is true or false, we can expand both sides and compare them to see if they are equal.

Expanding the left side:
C(A+B) = CA + CB

Expanding the right side:
AC + BC

Comparing the two sides:
CA + CB = AC + BC

Since matrix multiplication is not commutative, the order of addition matters. In this case, the left side has matrix C multiplied by matrix A and then added to matrix B, while the right side has matrix A multiplied by matrix C and then added to matrix B.

Therefore, the equation C(A+B) = AC + BC is false.

To check whether the equation C(A+B) = AC + BC is true or false, you need to verify it by substituting specific matrices for A, B, and C. Since you have matrices A, B, and C, all of which are 2x2, you can substitute them into the equation and perform the required calculations.

Let's assume the following matrices:
A = | a b |
| c d |

B = | e f |
| g h |

C = | i j |
| k l |

Now, let's substitute these matrices into the equation and perform the necessary calculations:

C(A+B) = C(A) + C(B) [Distributive property of matrix multiplication]

First, calculate A+B:
A + B = | a+e b+f |
| c+g d+h |

Next, calculate C(A):
C(A) = | i*a + j*c i*b + j*d |
| k*a + l*c k*b + l*d |

Similarly, calculate C(B):
C(B) = | i*e + j*g i*f + j*h |
| k*e + l*g k*f + l*h |

Now, let's compare both sides of the equation:

Left-hand side (LHS):
C(A+B) = | i*(a+e) + j*(c+g) i*(b+f) + j*(d+h) |
| k*(a+e) + l*(c+g) k*(b+f) + l*(d+h) |

Right-hand side (RHS):
AC + BC = | (i*a + j*c) + (i*e + j*g) (i*b + j*d) + (i*f + j*h) |
| (k*a + l*c) + (k*e + l*g) (k*b + l*d) + (k*f + l*h) |

By comparing the LHS and RHS, we can see that they are not the same. Therefore, the equation C(A+B) = AC + BC is False.

Regarding your statement, "Matrix multiplication is not commutative," that is correct. However, in this case, we are dealing with matrix addition, not multiplication, so the commutative property of matrix addition (A + B = B + A) doesn't apply.

So in summary:
- Your answer to the given equation is correct; it is False.
- Your statement about matrix multiplication not being commutative is also correct.
- Your statement about matrix addition being commutative is also correct.