Math Problem Statement

Укажите все верные утверждения о матрицах: 1) Все элементы скалярной матрицы равны между собой; 2) Произвольная диагональная матрица является симметрической матрицей; 3) Квадратная ступенчатая матрица является верхнетреугольной матрицей; 4) Все элементы на главной диагонали симметрической матрицы равны между собой.

Solution

The image contains four statements written in Russian, asking to select all the true ones regarding matrices. Here is the translation of these statements:

  1. All elements of a scalar matrix are equal to each other.
  2. An arbitrary diagonal matrix is a symmetric matrix.
  3. A square step-matrix (row echelon form) is an upper triangular matrix.
  4. All elements on the main diagonal of a symmetric matrix are equal to each other.

Let's analyze these statements:

  1. True. A scalar matrix is a diagonal matrix where all the diagonal elements are equal, so all elements of the matrix are the same.
  2. True. Any diagonal matrix is symmetric, as the transpose of a diagonal matrix is itself.
  3. True. A square matrix in row echelon form will have all zero entries below the diagonal, making it an upper triangular matrix.
  4. False. The elements on the diagonal of a symmetric matrix can be different. Symmetry only requires that the elements aij=ajia_{ij} = a_{ji}, but does not imply that the diagonal elements must be equal.

Therefore, the correct answers are the first three statements.

Would you like further details or clarifications?

Here are five related questions for more practice:

  1. What is the difference between a diagonal and a scalar matrix?
  2. Can a matrix be upper triangular but not in row echelon form?
  3. What makes a matrix symmetric?
  4. How do we check if a matrix is in row echelon form?
  5. What are the properties of a diagonal matrix?

Tip: A diagonal matrix is symmetric as long as all off-diagonal elements are zero.

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Math Problem Analysis

Mathematical Concepts

Matrix Theory
Symmetric Matrix
Diagonal Matrix
Upper Triangular Matrix

Formulas

-

Theorems

Properties of Scalar Matrix
Symmetry in Matrices

Suitable Grade Level

University Level - Linear Algebra