📘 linear algebra

1. **State the problem:** We are given a transformation defined by the matrix $$A = \begin{bmatrix}4 & 2 \\ -1 & 1\end{bmatrix}$$ and need to find its eigenvectors. 2. **Recall the

1. **State the problem:** Find the inverse of the matrix $$M = \begin{bmatrix}4 & 2 & 1 \\ -3 & 1 & 2 \\ 3 & 5 & 1\end{bmatrix}$$ using cofactors. 2. **Formula and rules:** The inv

1. The problem is to form a 3x3 matrix by placing the given values in the respective columns instead of rows. 2. A matrix is an array of numbers arranged in rows and columns. A 3x3

1. **State the problem:** Find the inverse of the matrix $$M = \begin{bmatrix}4 & -3 & 3 \\ 2 & 1 & 5 \\ 1 & 2 & 1\end{bmatrix}$$ using cofactors. 2. **Formula and rules:** The inv

1. **Problem statement:** Given the linear system: $$\begin{cases} x_1 + x_2 + x_3 = 1 \\ -x_1 + 2x_3 + x_4 = 4 \end{cases}$$

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

1. The problem is to understand and interpret the matrix $A$ given as: $$A=\begin{bmatrix}1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0\end{bmatrix}$$

1. The problem is to find a 3 by 3 matrix for which you can calculate its inverse. 2. A matrix must be square (same number of rows and columns) and have a non-zero determinant to h

1. **Problem Statement:** Find the limit as $x \to \infty$ of the product of nine identical $2 \times 2$ matrices:

1. Let's create a practice problem involving pivot rows and columns in a matrix. 2. Problem: Given the matrix $$\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}$$, p

1. Let's start by stating the problem: Understanding what pivot rows and pivot columns are in the context of matrices and linear algebra. 2. In linear algebra, when performing Gaus

1. Let's start by understanding what pivot rows and pivot columns are in the context of a matrix. 2. A pivot position in a matrix is the first nonzero entry in a row after the matr

1. The problem involves analyzing a 9x9 grid with given cell values and corresponding row and column sums. 2. We want to verify the consistency of the sums and understand the distr

1. The problem involves understanding when to apply a certain method to a system of equations or conditions. 2. A system is called consistent if it has at least one solution.

1. The problem: Understand when to apply least squares approximation in matrices. 2. Least squares approximation is used when you have an overdetermined system of linear equations,

1. **Problem statement:** Given the matrix $$A = \begin{bmatrix} i & 0 \\ 1 & -i \end{bmatrix},$$ show that $$A^4 = I_2,$$ where $$I_2$$ is the 2x2 identity matrix. 2. **Recall:**

1. **Problem Statement:** Given the matrix $A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}$, show that $A^4 = I_2$, where $I_2$ is the $2 \times 2$ identity matrix. 2. **Recall

1. **Problem Statement:** Q1: Given matrices $A$, $B$, and $C$ such that $A + B = C$, find $a, b, c, d$ (elements of matrices) and then find $AB$ and $CB$ after substitution.

1. Let's start by defining what a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. 2. A rectangular matrix is any matrix where the number of rows

1. Let's start by stating the problem: Understanding what a matrix is and how it works. 2. A matrix is a rectangular array of numbers arranged in rows and columns. For example, a m

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) and its properties. 2. A matrix is a rectangular array of numbers arranged in rows and col