📘 linear algebra

1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 2 \\ 1 & 1 & 2 \end{bmatrix}$$ using the adjoint method. 2. **Formula and Impor

1. **Problem 1: Matrix operations with 2x2 matrices** Given matrices:

1. مسئله: معکوس ماتریس \( A = \begin{bmatrix} 2 & 2 & 2 \\ 1 & 1 & 1 \\ -1 & 1 & 1 \end{bmatrix} \) را بیابید. 2. فرمول: معکوس ماتریس \( A \) زمانی وجود دارد که دترمینان آن غیر صفر

1. مسئله: معکوس ماتریس‌های داده شده را بیابید. 2. فرمول و نکات مهم:

1. **State the problem:** We are given the matrix $$\begin{bmatrix} 1 & -2 & 3 & 1 \\ 2 & -1 & 2 & 2 \\ 3 & 1 & 2 & 3 \end{bmatrix}$$

1. The problem asks why minus signs appear outside the matrix elements when calculating the adjoint (adjugate) of matrix $A$. 2. The adjoint matrix is the transpose of the cofactor

1. **Problem:** Given $$ A = \begin{bmatrix} a & a & 0 \\ a & 1 & 0 \\ 0 & 1 & 2 \end{bmatrix} $$

1. **Problem:** Given matrix $$A=\begin{bmatrix}a & a & 0 \\ a & 1 & 0 \\ 0 & 1 & 2\end{bmatrix}$$ where $a$ satisfies $x^2 - x + 1 = 0$, find

1. The problem describes a cyclic tridiagonal matrix $A$ defined by entries involving $\lambda_i$ and $\mu_i$ which depend on the mesh sizes $h_i$ of a partition $a = x_0 < x_1 < \

1. Problem statement. Apply Crout LU to the matrix $A_{2b}$ and solve $A_{2b}x=b_{2b}$ via $L_{2b}y=b_{2b}$ and $U_{2b}x=y$, and state pivot checks.

1. The user requests a textbook outline for Higher Mathematics for engineers covering 15 broad topics. 2. Since the request is for a full textbook preparation, which is extensive,

1. The problem is to understand the definition of a matrix and see some examples. 2. A matrix is a rectangular array of numbers arranged in rows and columns. It is usually denoted

1. **Problem Statement:** Calculate the characteristic equation of matrix a): $$\begin{bmatrix} 1 & 2 & 3 \\ 0 & -2 & 6 \\ 0 & 0 & -3 \end{bmatrix}$$

1. **Problem Statement:** Find the rank of each matrix (a, b, c) using the echelon method. 2. **Formula and Rules:** The rank of a matrix is the number of nonzero rows in its row e

1. The problem is to analyze the matrix equation $$A^3 - (A^2 - A) = 0$$ for a matrix $A \in M_n(\mathbb{R})$ with rank $\mathrm{rk}(A) = M$ and to deduce properties about $A$ and

1. **State the problem:** We need to transform the matrix $$A=\begin{pmatrix} -\frac{5}{2} & -\frac{7}{4} & -5 & \frac{35}{3} \\ -1 & \frac{3}{4} & -2 & 3 \\ \frac{1}{4} & -2 & \fr

1. **Problem Statement:** Given square invertible matrices $A$, $B$, and $C$ of the same size, with $B^T B = I = B B^T$ and $C$ having no eigenvalue equal to $-1$, find the inverse

1. **Problem Statement:** Given square invertible matrices $A$, $B$, and $C$ of the same size, with $B^T B = I = B B^T$ and $C$ having no eigenvalue equal to $-1$, find the inverse

1. State the problem: Solve the system of linear equations using Gauss's Method: $$\begin{cases} 2x + 3y - z = 5 \\ 4x + y + 2z = 6 \\ -2x + 5y + 3z = 4 \end{cases}$$

1. Statement of the problem: Multiply the scalar 1 by the matrix $$\begin{bmatrix}a & b\\ c & d\end{bmatrix}$$. 2. Formula used: For a scalar k and matrix M the scalar multiplicati

1. **Problem:** Find the product of matrices $A$ and $B$ where $$A=\begin{pmatrix}1 & 2 & 0 & 3 \\ 2 & 2 & 6 & -9 \\ 1 & -1 & 2 & 1\end{pmatrix},\quad B=\begin{pmatrix}2 & 2 & 5 \\