📘 linear algebra

1. Problem: Find eigenvalues of $$A''$$ given eigenvalues of $$A = \begin{pmatrix}3 & -1 & 1 \\ -1 & 5 & -1 \\ 1 & -1 & 3\end{pmatrix}$$ are 3, 6, and the matrix is 3x3. Step 1: Gi

1. The problem is to express the vector \(\overline{x}\) as a linear combination of vectors \(\overline{p}, \overline{q}, \overline{r}\), i.e., find scalars \(a, b, c\) such that $

1. Solve the system \(\{x_1 + x_2 + 2x_3 = 8, -x_1 - 2x_2 + 3x_3 = 1, 3x_1 - 7x_2 + 4x_3 = 10\}\) by Gaussian elimination. Step 1: Write augmented matrix:\

1. We are given augmented matrices in row echelon form and need to identify pivot rows and columns and solve the system for each part. 2. Part (a):

1. The cross product, mathematically defined, is an operation between two vectors in three-dimensional space, producing a vector that is perpendicular to both. 2. In 3D, for two ve

1. The user seems to input a matrix with elements \(a, b, c, d\).\n2. Since the input is incomplete or malformed (\texttt{\begin{bmatrix}a&b\\c&d\end{bmatpoly}), it looks like the

1. Given matrices $$A=\begin{bmatrix}-1 & 1 & -1 \\ -3 & -2 & 0 \\ -2 & 1 & 0\end{bmatrix},

1. **Énoncé du problème :** Calculer les matrices $F = A + B + C$, $G = A - B - C$, puis $H = tA + tB - tC$, où $tM$ désigne la transposée d'une matrice $M$. Matrices données : $$A

**Problem statement:** Three sectors A (Agriculture), U (Urban), I (Industry) consume and supply water interdependently with given usage coefficients. Let $p_1, p_2, p_3$ be prices

1. Рассмотрим понятия. 2. Сокращение рядов — это процесс уменьшения размера матрицы, путем удаления строк или столбцов, не влияющих на ранг матрицы или на свойства решения системы.

1. **Problem Statement:** Given matrix $$ A = \begin{bmatrix}1 & 2 & 3 & 4 \\

1. **State the problem:** You have a 3x3 matrix $$A = \begin{bmatrix} 1 & 0 & -2 \\ 4 & 2 & 7 \\ 1 & -5 & 4 \end{bmatrix}$$. 2. Since the problem provides the matrix $A$ but does n

1. The problem is to understand the matrix $A$ provided: $$A = \begin{bmatrix} 1 & 0 & -2 \\ 4 & 2 & 7 \\ 1 & -5 & 4 \end{bmatrix}$$

1. The problem asks us to find vectors parallel to the vector $$\begin{pmatrix}-4 \\ 2\end{pmatrix}$$. 2. Two vectors are parallel if one is a scalar multiple of the other.

1. Muammo: Berilgan matritsalarning rangini aniqlash: a) minorlar usulida, b) nol yig’ish usulida.

1. Problem: Find the ranks of matrices A (3x5) using minor method and matrices B (4x4) using zero sum method for all given pairs. 2. Minor Method (A matrices): The rank is the size

1. Problem (a): Find the SVD of $A = \begin{pmatrix} -2 & 2 \\ 1 & 1 \end{pmatrix}$. Step 1: Compute $A^TA$ and $AA^T$:

1. **State the problem:** We need to find the dimension of the subspace $S = \{(2a + 4b - c,\ 3a - 6b,\ -4a + b + 3c,\ -a + 3c) : a,b,c \in \mathbb{R}\}$, which is a subset of $\ma

1. **State the problem:** We are given a matrix $A = \begin{bmatrix}-1 & 5 & 7 \\ 0 & -2 & 4\end{bmatrix}$, a vector $b = \begin{bmatrix}9 \\ 6\end{bmatrix}$, and a vector $x = \be

**Exercice 1** Soit la matrice $$A = \begin{pmatrix}0 & 2 & -1 \\ 3 & -2 & 0 \\ -2 & 2 & 1\end{pmatrix}$$.

1. Problem No. 2: Solve the system of linear equations using the Gauss-Jordan method: $$\begin{cases} -x_1 + 4x_2 + x_3 = 8 \\ \frac{5}{3}x_1 + \frac{2}{3}x_2 + \frac{2}{3}x_3 = 1