Final answer:
The statement (a + b)ᵀ = aᵀ + bᵀ is false for matrices. The statement is False; the transpose of the sum of two matrices is equal to the sum of their transposes. This showcases the commutative property of matrix addition, similar to the addition of real numbers.
Step-by-step explanation:
False
The statement (a + b)ᵀ = aᵀ + bᵀ is false for matrices.
For matrices, the transpose of the sum of two matrices is not equal to the sum of their transposes.
Therefore, the correct answer is false.
The statement is true; the transpose of the sum of two matrices is equal to the sum of their transposes. This showcases the commutative property of matrix addition, similar to the addition of real numbers.
If a and b are the same size matrix, it is indeed true that (a + b)ᵀ = aᵀ + bᵀ. This property reflects the fact that matrix addition is commutative, similar to the addition of ordinary numbers where the order does not affect the result. For instance, just as adding 2 + 3 gives the same result as 3 + 2, matrix addition follows the same rule.
Furthermore, the transpose operation has the property that when you transpose the sum of two matrices, it is the same as summing the transposes of these matrices individually. Mathematically put, if 'A' and 'B' are matrices, then (A + B)ᵀ = Aᵀ + Bᵀ, which is an application of the transpose operation on the commutative property of matrix addition.