📘 linear algebra

1. **Problem Statement:** Given matrices $$A = \begin{bmatrix}-1 & 2 & 0 \\ 3 & -5 & 1 \\ -1 & 2 & 0\end{bmatrix}$$

1. **Problem Statement:** Find the product of two 3x3 matrices:

1. **Problem statement:** Find the inverse of the matrix $$A = \begin{bmatrix} 2 & 1 & 3 \\ 3 & 0 & 1 \end{bmatrix}$$ 2. **Important note:** The inverse of a matrix exists only if

1. The problem asks why the addition of matrix A and matrix B is not possible. 2. Matrix A is a 2x2 matrix: $$A = \begin{bmatrix}6 & 6 \\ 5 & 2\end{bmatrix}$$

1. **Stating the problem:** We are given three matrices multiplied together and asked to find the final result vector $\mathbf{x} = (x_1, x_2, x_3, x_4)$ by performing Gaussian eli

1. **Problem statement:** Find the matrix of the linear operator $L : \mathcal{M}_{2,2}(\mathbb{R}) \to \mathcal{M}_{2,2}(\mathbb{R})$ defined by $$L \begin{pmatrix} x & y \\ z & w

1. **Problem:** Find the inverse of matrix $$A = \begin{pmatrix}1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8\end{pmatrix}$$. 2. **Formula and rules:** The inverse of a matrix $$A$$, denoted

1. **Problem:** Find the rank of matrix $$A = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & -4 \\ 0 & 4 & 0 \end{bmatrix}$$. 2. **Recall:** The rank of a matrix is the maximum number of li

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. This is a 2x2 matrix with elements \(a, b, c, d\) arranged in two rows and two columns

1. **State the problem:** Find the inverse of matrix $A$ using the Cayley–Hamilton theorem, where $$A=\begin{bmatrix}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2\end{bmatrix}$$

1. **Stating the problem:** We have two matrices:

1. **State the problem:** Given matrix $$A = \begin{pmatrix}34 & 26 & 20 \\ -68 & -51 & -38 \\ 32 & 23 & 16\end{pmatrix}$$, find the eigenvalues and eigenvectors of $$A^3$$ without

1. **Énoncé du problème :** Définir et expliquer les notions de scalaire, vecteur, matrice, tenseur, transposition, matrice identité, inverse de matrice, ainsi que les opérations d

1. **Problem statement:** Solve the system of equations $$\begin{cases} x + 4y + z = 1 \\ 8x + 64y - 2z = 12 \\ 9x + 154y + 9z = 12 \end{cases}$$

1. The problem is to determine if Cramer's rule can be applied to a given system of linear equations. 2. Cramer's rule applies to a system of $n$ linear equations with $n$ unknowns

1. **Problem statement:** Given matrix $$A = \begin{pmatrix}1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1\end{pmatrix}$$, show that $$A^2 = 4A + 5I$$ where $$I$$ is the 3x3 identity matrix.

1. **State the problem:** Solve the system of linear equations given by $$\begin{pmatrix} 3 & 2 & 1 \\ 2 & 3 & 2 \\ 1 & 2 & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{p

1. Muammo: Berilgan matritsalar $A$ va $B$ uchun ifodani hisoblang: $$-2A + 7B$$. 2. Formulalar va qoidalar: Matritsalar ustida chiziqli amallar quyidagicha bajariladi:

1. Muammo: Quyidagi matritsalar ustida chiziqli amallarni bajarish. 2. Chiziqli amallar - bu matritsalar ustida bajariladigan qo'shish va skalyar ko'paytirish amallari.

1. **State the problem:** We are given two matrices $$A = \begin{pmatrix} -1 & 0 & -6 \\ 2 & 1 & 4 \\ 5 & 8 & -9 \end{pmatrix}$$

1. **Problem Statement:** Decode the message transmitted by the matrix $$M' = \begin{pmatrix} 51 & 40 & 36 & 51 & 37 & 42 \\ 85 & 81 & 71 & 114 & 64 & 84 \\ 126 & 130 & 113 & 196 &