📘 Linear Algebra

1. **Problem statement:** We have a transformation represented by the matrix $$\begin{bmatrix}1 & -1 \\ 0 & 2\end{bmatrix}$$.

1. The problem is to understand the transformation represented by a given matrix. 2. A transformation matrix is used to perform linear transformations such as rotations, scalings,

1. **State the problem:** We need to find the value of $x$ for which the matrix $$\begin{pmatrix}8 & x & 0 \\ 4 & 0 & 2 \\ 12 & 6 & 0\end{pmatrix}$$

1. **Problem statement:** We are given a function $f(x)$ defined by a $4\times4$ matrix with entries involving $x$ and constants. We want to find the constant term of $f(x)$, which

1. The problem asks: "What's the matrix here?" but no specific matrix or context is provided. 2. A matrix is a rectangular array of numbers arranged in rows and columns.

1. **Problem Statement:** Given an $n \times n$ matrix $M$ with real entries such that $M^3 = I$ and $Mv \neq v$ for any nonzero vector $v$, determine which statements (A) to (D) a

1. **Problem Statement:** Given distinct non-zero real numbers $a, b, c, d$ such that $a + b = c + d$, find an eigenvalue of the matrix

1. **State the problem:** We are given a matrix

1. **Problem:** Find the eigenvalues and eigenvectors of matrix \( A = \begin{bmatrix}4 & 6 & 6 \\ 1 & 3 & 2 \\ -1 & -4 & -3\end{bmatrix} \). 2. **Formula:** Eigenvalues \( \lambda

1. **Problem statement:** Given two vectors $\vec{a}, \vec{b} \in \mathbb{R}^2$, determine which statements about vectors and their scalar product are true. 2. **Recall the scalar

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

1. **Problem Statement:** Find the unique fixed probability vector $\mathbf{p}$ of the stochastic matrix $$AB=\begin{bmatrix}1 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 4 & 1 \\ 6 & 1 & 3 \\ 1 &

1. **Problem Statement:** Find the unique fixed probability vector $\mathbf{p}$ of the stochastic matrix $$AB=\begin{bmatrix}1 & 2 & 1 \\ 2 & 1 & 3 \\ 1 & 4 & 1 \\ 6 & 1 & 3 \\ 1 &

1. **State the problem:** We want to find all solutions of a linear system whose augmented matrix can be row-reduced to $$\begin{bmatrix}1 & 0 & 1 & 4 \\ -1 & 2 & 3 & 2\end{bmatrix

1. **State the problem:** We are given a 3×3 matrix $$A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$$ and need to determine if it is idempotent. A matr

1. **Problem 1: Find the rank of matrix A** Matrix A is given by:

1. **Problem 1: Find the rank of matrix** Given matrix $$A=\begin{bmatrix}2 & 4 & 1 & 3 \\ 1 & 2 & 1 & 2 \\ 4 & 8 & 3 & 7 \\ 0 & 0 & -1 & -1\end{bmatrix}$$

1. **Problem:** Check if $\mathbf{v} = (3, 2, 1)$ can be written as a linear combination of $\mathbf{u}_1 = (1, 0, 1)$ and $\mathbf{u}_2 = (0, 1, 1)$. **Step:** We want to find sca

1. **Problem Statement:** Analyze whether each given 3x3 matrix $A$ is diagonalizable. If yes, find the diagonal matrix $D$ and the eigenvector matrix $P$ such that $$P^{-1} A P =

1. **State the problem:** We are given the matrix $$\begin{bmatrix}-2 & -1 & -3 & 0 \\ 2 & 0 & 0 & 0\end{bmatrix}$$

1. **State the problem:** Find all values of $k$ such that the matrix-vector products equal the zero vector in the given equations.