The task asks to select all true statements about matrices. Let's go through each statement one by one:

1. $(A + B)^T = A^T + B^T$ for arbitrary matrices $A$ and $B$:

   • This is true. The transpose of the sum of two matrices is the sum of their transposes.

2. $(\lambda A)^T = \lambda A^T$ for arbitrary matrix $A$ and scalar $\lambda$:

   • This is true. The scalar multiplication of a matrix remains unchanged under transposition.

3. All elements of a scalar matrix are equal to each other:

   • This is false. A scalar matrix has the same scalar on the diagonal, but the off-diagonal elements are zeros.

4. A transpose of an upper triangular matrix becomes a lower triangular matrix:

   • This is true. The transpose of an upper triangular matrix flips it to become a lower triangular matrix.

Thus, the correct answers are:

• First, second, and fourth statements are true.

Would you like further details or have any questions?

Here are 5 related questions to explore:

1. What are the properties of matrix transposition?
2. How does scalar multiplication affect matrix transposition?
3. What is the difference between a scalar and a diagonal matrix?
4. How do triangular matrices behave under transposition?
5. Can an upper triangular matrix ever be symmetric?

Tip: Remember that matrix operations often preserve or reflect certain properties, like symmetry and triangular structure.