Let's analyze each of the statements in the image and determine their correctness:

1. "Квадратная ступенчатая матрица является верхнетреугольной матрицей."

   • Incorrect. A square step matrix (ступенчатая) does not necessarily have zeros below the diagonal as required for an upper triangular matrix (верхнетреугольная). Therefore, not every square step matrix is upper triangular.

2. "Все элементы скалярной матрицы равны между собой."

   • Correct. In a scalar matrix, all diagonal elements are equal, and all off-diagonal elements are zero. Therefore, the statement holds true.

3. "Все элементы на главной диагонали симметрической матрицы равны между собой."

   • Incorrect. In a symmetric matrix, the elements are symmetric about the diagonal, but there is no requirement that all diagonal elements are equal. They can differ.

4. "Произвольная диагональная матрица является симметрической матрицей."

   • Correct. A diagonal matrix is symmetric because all off-diagonal elements are zero, and symmetry only requires that $a_{ij} = a_{ji}$, which is true for diagonal matrices.

Correct Answers:

• Statement 2 (about scalar matrices).
• Statement 4 (about diagonal matrices).

Do you need further clarification on any of the statements? Let me know!

Related Questions:

1. What is the definition of an upper triangular matrix?
2. How does a scalar matrix differ from a diagonal matrix?
3. Can a symmetric matrix have off-diagonal elements that are nonzero?
4. What are the key properties of a step (row-echelon) matrix?
5. What is the relationship between square and diagonal matrices?

Tip:

In a symmetric matrix, the condition $a_{ij} = a_{ji}$ applies to all elements, so symmetry does not impose conditions on diagonal elements alone.