Let's analyze the statements in the image to determine which are correct.

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

   • A scalar matrix is a diagonal matrix where all diagonal elements are equal, and the non-diagonal elements are zero. However, this statement says that all elements are equal, which is incorrect because only the diagonal elements are equal in a scalar matrix, not the entire matrix. Hence, this statement is false.

2. "Верхнетреугольная матрица при транспонировании становится нижнетреугольной матрицей."

   • This is correct. When you transpose an upper triangular matrix (a matrix where all elements below the diagonal are zero), it becomes a lower triangular matrix (where all elements above the diagonal are zero). Hence, this statement is true.

3. "Для произвольных матриц $A$ и $B$ справедливо равенство $(A + B)^T = A^T + B^T$."

   • This is a standard property of matrix transposition. The transpose of the sum of two matrices is the sum of their transposes. Hence, this statement is true.

4. "Для произвольной матрицы $A$ справедливо равенство $(\lambda A)^T = \lambda A^T$."

   • This is also true. The transpose of a matrix scaled by a scalar $\lambda$ is the same as scaling the transpose of the matrix by $\lambda$. Hence, this statement is true.

Conclusion:

The correct statements are:

• "Верхнетреугольная матрица при транспонировании становится нижнетреугольной матрицей."
• "Для произвольных матриц $A$ и $B$ справедливо равенство $(A + B)^T = A^T + B^T$."
• "Для произвольной матрицы $A$ справедливо равенство $(\lambda A)^T = \lambda A^T$."

Would you like more details or have any other questions?

Here are 5 related questions you can explore:

1. What is the difference between a scalar matrix and a diagonal matrix?
2. How is the transpose of a matrix calculated for higher-dimensional matrices?
3. What other properties hold for matrix transposition?
4. How does transposing affect special matrices like symmetric or orthogonal matrices?
5. Can triangular matrices be used to solve systems of linear equations?

Tip: Matrix transposition has the useful property of preserving addition and scalar multiplication, making it a powerful tool for matrix algebra!