📘 Linear Algebra

1. **State the problem:** Find the inverse of the matrix $$A = \begin{pmatrix} 2 & -1 & 3 \\ 4 & 0 & 1 \\ -2 & 5 & 2 \end{pmatrix}$$ using the adjoint method. 2. **Formula and rule

1. **Problem Statement:** Find the determinant of the matrix $$A = \begin{bmatrix} 1 & 0 & 0 & -1 \\ 3 & 1 & 2 & 2 \\ 1 & 0 & -2 & 1 \\ 2 & 0 & 0 & 1 \end{bmatrix}$$ using cofactor

1. **State the problem:** We need to find the determinant of the matrix $$A = \begin{bmatrix} 0 & 1 & 5 \\ 3 & -6 & 9 \\ 2 & 6 & 1 \end{bmatrix}$$ using row reduction. 2. **Recall

1. The problem is to find the determinant of the matrix $$A = \begin{bmatrix} 2 & 4 & 1 \\ 5 & 2 & 3 \\ 1 & 4 & 8 \end{bmatrix}$$. 2. The formula for the determinant of a 3x3 matri

1. **State the problem:** We are given a set of vectors \( M = \{(1,0,1,0), (1,1,1,1), (-1,2,0,1)\} \) and asked to orthogonalize this basis using the Gram-Schmidt process. 2. **Re

1. **Problem statement:** Given vectors $$u_1=\begin{bmatrix}1\\2\\0\end{bmatrix}, u_2=\begin{bmatrix}2\\1\\1\end{bmatrix}, u_3=\begin{bmatrix}3\\3\\1\end{bmatrix}, u_4=\begin{bmat

1. **Problem statement:** We need to set up and solve the system of linear equations representing the traffic flow in the city network to find the values of $x_1, x_2, x_3, x_4, x_

1. **Problem Statement:** Find the rank of the matrix $$A = \begin{bmatrix} 0 & -11 & -4 \\ 2 & 6 & 2 \\ 4 & 1 & 0 \end{bmatrix}$$. 2. **Recall:** The rank of a matrix is the maxim

1. **Problem Statement:** Find the inverse of matrix 21 and verify that $M^{-1}M=I$. Matrix 21 is:

1. **Problem statement:** Given matrix $A = \begin{pmatrix} 2 & 7 \\ 4 & k \end{pmatrix}$ and an eigenvector $V_1 = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$, find the value of $k$ and

1. Let's start by understanding what a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. 2. The size of a matrix is given by the number of rows an

1. **Problem Statement:** Given matrices $$A = \begin{pmatrix} 1 & -1 \\ 2 & -1 \end{pmatrix}, \quad B = \begin{pmatrix} a & 1 \\ b & -1 \end{pmatrix}$$

1. **Problem statement:** Given vectors $u, v_1, \ldots, v_k$ in $\mathbb{R}^n$ such that the set $\{v_1, \ldots, v_k\}$ spans $\mathbb{R}^n$. Consider the equations: (1) $x_1 v_1

1. **Problem 4: Solve the system using Jacobi's method with initial guess $x^{(0)} = 0$ up to two iterations.** Given system:

1. Let's start by understanding what a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. For example, a matrix $A$ with 2 rows and 3 columns looks

1. **State the problem:** We are given vectors $x = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$, $y = \begin{pmatrix} 5 \\ -2 \end{pmatrix}$, and $z = \begin{pmatrix} -4 \\ 13 \end{pmatr

1. **State the problem:** Given vectors $x = \begin{bmatrix}2 \\ 3\end{bmatrix}$, $y = \begin{bmatrix}5 \\ -2\end{bmatrix}$, and $z = \begin{bmatrix}-4 \\ 13\end{bmatrix}$, find sc

1. **Stating the problem:** We want to understand the dot product of two vectors and its properties. 2. **Definition:** The dot product (also called scalar product) of two vectors

1. **Stating the problem:** We want to understand the dot product of two vectors and its properties. 2. **Definition:** The dot product of two vectors $\mathbf{a} = (a_1, a_2, \ldo

1. The problem is to find the vector projection of vector \(\mathbf{a}\) onto vector \(\mathbf{b}\) without using the orthogonal projection method. 2. The formula for the projectio

1. **Problem Statement:** Given matrices: