📘 linear algebra

1. The problem is to find the inverse of the matrix $$\begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}$$. 2. The formula for the inverse of a 2x2 matrix $$\begin{bmatrix}a & b \\ c & d\e

1. **Stating the problem:** We have a transformation matrix $$\begin{bmatrix} b & -2 \\ 16 & c \end{bmatrix}$$ that maps the point $$\begin{bmatrix} 5 \\ 1 \end{bmatrix}$$ onto the

1. **Problem Statement:** Given a matrix $A$ with two eigenvalues equal to 1 each, find the eigenvalues of the matrix $5A$. 2. **Formula and Rules:**

1. **Problem Statement:** Find the eigenvalues and corresponding eigenvectors of the matrix $$A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{bmatrix}$$

1. The problem is to express a system or equation in matrix form. 2. The matrix form of a system of linear equations is generally written as $$AX = B$$ where:

1. **Problem:** Define a Vector Space and verify if the set of all 3 × 3 upper triangular matrices with determinant 0 forms a vector space under standard matrix addition and scalar

1. The problem is to understand the structure of the matrix $R$ given as: $$R = \begin{bmatrix} r_{xx} & r_{yx} & r_{zx} \\ r_{yy} & r_{zy} \\ r_{zz} \end{bmatrix}$$

1. مسئله: حل یک ماتریس 3 در 3 به روش مودال (Modal Method) است. 2. روش مودال چیست؟

1. **State the problem:** Calculate the determinant of the given 5x5 matrix: $$\begin{bmatrix} 1 & 2 & 3 & 1 & 5 \\ 0 & 1 & 0 & 5 & 1 \\ 2 & 1 & 2 & 3 & 2 \\ 0 & 3 & 0 & 1 & 3 \\ 3

1. The problem asks to find $3A$ where $A$ is the matrix $\begin{bmatrix} 2 & -4 \\ 1 & 0 \end{bmatrix}$. 2. The formula for scalar multiplication of a matrix is $kA = k \times A$,

1. **Problem statement:** Find the area of the triangle with vertices $A(1,-2,2)$, $B(5,-6,2)$, and $C(1,3,-1)$, and find the height $h$ from point $B$ to the line segment $AC$. 2.

1. **State the problem:** Find the point $P$ on line segment $AC$ such that the height $h$ from point $B$ to $AC$ is perpendicular to $AC$. 2. **Formula and rules:** The height fro

1. Problem: Find $(RS)^{-1}$ given matrices $$R=\begin{bmatrix}3 & -5 \\ 3 & 9\end{bmatrix}, \quad S^{-1}=\begin{bmatrix}-1 & -2 \\ 5 & 3\end{bmatrix}$$

1. **State the problem:** We are given two matrices $A = \begin{bmatrix} 5 & -1 \\ 0 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 3 \\ 4 & -2 \end{bmatrix}$ and need to find the

1. **State the problem:** We are given the matrix $$A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$$ and need to find its transpose. 2. **Formula for transpose:** The transpose o

1. **State the problem:** Find the eigenvalues of the matrix $$\begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}$$. 2. **Formula used:** Eigenvalues $\lambda$ satisfy the characteristic

1. **Problem Statement:** Determine the rank of the matrix $$\begin{bmatrix}3 & 2 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}$$. 2. **Definition:** The rank of a matrix is the maxim

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$. 2. **Formula for Inverse of a 2x2 Matrix:** For a matrix $$A = \begin

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix