📘 linear algebra

1. Statement of the problem. Determine which one of the following properties does not hold for matrix multiplication.

1. **Problem:** Find the rank of matrix \( A = \begin{bmatrix} 3 & 4 & 1 & 1 \\ 2 & 4 & 3 & 6 \\ -1 & -2 & 6 & 4 \\ 1 & -1 & 2 & -3 \end{bmatrix} \) using row echelon form. 2. **Fo

1. **Problem Statement:** Find the eigenvalues and eigenvectors of the matrix $$A = \begin{bmatrix} 2 & 1 & 1 \\ 2 & 1 & -2 \\ -1 & 0 & -2 \end{bmatrix}$$. 2. **Formula and Explana

1. **Problem:** Find the angle between vectors $\vec{a} = (3,0,4)$ and $\vec{b} = (5,1,-1)$. 2. **Formula:** The angle $\theta$ between two vectors $\vec{a}$ and $\vec{b}$ is given

1. Асуудал: Матрицын урвууг олох. Матриц өгөгдсөн: $$A = \begin{pmatrix} 3 & 2 & -1 \\ 4 & 5 & 2 \\ -2 & 1 & 4 \end{pmatrix}$$

1. Задача: Найти обратную матрицу для матрицы $$A=\begin{pmatrix}1 & 0 & N \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$$. 2. Формула для обратной матрицы: $$A^{-1} = \frac{1}{\det(A)}

1. **State the problem:** Solve the system of linear equations using Gaussian elimination, scaling, and pivoting: $$\begin{cases} 2x_2 + x_4 = 0 \\ 2x_1 + 2x_2 + 3x_3 + 2x_4 = -2 \

1. **Determine $\lambda$ and $\mu$ for many solutions:** Given the system:

1. **Problem Statement:** Learn about matrices, their types, and examples including diagonal, inverse, identity, and determinant. 2. **Definition:** A matrix is a rectangular array

1. **Problem Statement:** Show that the linear operator $T$ on $\mathbb{R}^3$ represented by the symmetric matrix $$

1. Muammo: Berilgan tenglama sistemasini yeching: $$\begin{cases} a = x_1 + x_2 + 2x_3 \\ b = -1 \\ X = A^{-1}B \end{cases}$$

1. **Problem Statement:** Find the Reduced Row Echelon Form (RREF) of the matrix $$\begin{bmatrix}1 & 2 & -1 & 8 \\ -3 & -6 & 2 & -11 \\ 2 & 4 & 1 & 7\end{bmatrix}$$. 2. **Formula

1. **Problem Statement:** Find the Reduced Row Echelon Form (RREF) of the matrix $$\begin{bmatrix}1 & 2 & 1 & 4 \\ 2 & 4 & -3 & 1 \\ 3 & 6 & -5 & 2\end{bmatrix}$$. 2. **Formula and

1. **State the problem:** Determine which pairs of the given planes are orthogonal (perpendicular) and which are parallel. 2. **Recall the formula and rules:**

1. **State the problem:** Find matrix $X$ such that $$A \cdot X \cdot C + 2 \cdot B = C$$ where $$A = \begin{bmatrix} 2 & 1 \\ -1 & 3 \end{bmatrix}, B = \begin{bmatrix} 2 & -1 \\ 1

1. مسئله: مقدار $m$ را بیابید به طوری که دو بردار $$\vec{b} = (m -1) \hat{i} - \hat{j} + r \hat{k}$$

1. The problem is to identify the correct vector option among A, B, C, and D. 2. Since no specific question or context is given (such as a vector equation, transformation, or condi

1. **Problem statement:** Determine which subsets of $\text{Mat}_2(\mathbb{R})$ are subspaces. 2. **Recall:** A subset $W$ of a vector space $V$ is a subspace if it satisfies:

1. **Problem statement:** Given the vector $v = \begin{bmatrix}-1 \\ 1 \\ 2\end{bmatrix}$, compute the Householder matrix $H_v$ and determine the effect of multiplying $H_v$ by $v$

1. **Problem statement:** Given the vector $v = \begin{bmatrix}-1 \\ 1 \\ 2\end{bmatrix}$, compute the Householder matrix $H_v$ and determine the effect of multiplying $H_v$ by $v$

1. **State the problem:** Find the determinant and inverse of the matrix $$A = \begin{pmatrix} 2 & 0 & 1 & -13 \\ 2 & 0 & 40 & -1 \\ 5 & 2 & 1 & 0 \\ 3 & 0 & 0 & 0 \end{pmatrix}$$