📘 linear algebra

1. The problem provides several matrices and a quadratic form $q(x,y,z) = 3x^2 + 4xy - y^2 + 8xz - 6yz + z^2$. We will analyze each given matrix and the quadratic form step-by-step

1. **Problem Statement:** We are given matrices and vectors and need to perform matrix operations: scalar multiplication, addition, subtraction, and matrix multiplication.

1. **State the problem:** Find the determinant of the 4x4 matrix $$\begin{bmatrix} 3 & 3 & 0 & 5 \\ 2 & 2 & 0 & -2 \\ 4 & 1 & -3 & 0 \\ 2 & 10 & 3 & 2 \end{bmatrix}$$

1. The problem is to find the determinant of the matrix: $$\begin{bmatrix} 3 & 3 & 0 & 5 \\ 2 & 2 & 0 & -2 \\ 4 & 1 & -3 & 0 \\ 2 & 10 & 3 & 2 \end{bmatrix}$$

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. This is a 2x2 matrix with elements \(a, b, c, d\) arranged as:

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

1. **State the problem:** Solve the system of linear equations given by the augmented matrix: $$\begin{bmatrix} 1 & 3 & -2 & 0 & 2 & 0 & | & 0 \\ 2 & 6 & -5 & -2 & 4 & -3 & | & -1

1. **Solve the system using Gauss elimination:** Given:

1. The problem is to find the matrix $2 \times A$ where $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & 1 \\ 0 & 0 & 4 \end{bmatrix}$. 2. To multiply a matrix by a scalar, multiply each

1. The problem is to find the square of the matrix $$A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & 1 \\ 0 & 0 & 4 \end{bmatrix}$$, i.e., compute $$A^2 = A \times A$$. 2. Multiply matri

1. The problem is to find $A^2$, which means multiplying matrix $A$ by itself. 2. Suppose matrix $A$ is given by $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$.

1. **State the problem:** Given matrix $A = \begin{bmatrix}1 & -1 & 2 \\ 0 & 2 & 1 \\ 0 & 0 & 4\end{bmatrix}$, find matrix $B = A^2 - 2A + 3I$, where $I$ is the $3 \times 3$ identi

1. The problem is to perform LU factorization of a given square matrix $A$. 2. LU factorization decomposes matrix $A$ into the product of a lower triangular matrix $L$ and an upper

1. **State the problem:** We need to find $\left(I - M\right)^{-1} D$ where $$M = \begin{bmatrix}0.4 & 0.1 & 0.0 \\ 0.2 & 0.6 & 0.1 \\ 0.1 & 0.1 & 0.4\end{bmatrix}, \quad D = \begi

1. The problem is to understand how to maintain the order of sectors when forming matrices. 2. When forming matrices from sectors, the order of sectors corresponds to the order of

1. **State the problem:** 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. **Calculate the determi

1. The problem states that $x_{opt}$ is the least squares solution to $Ax = b$ and asks about the nature of the error vector $e = Ax_{opt} - b$. 2. Recall that the least squares so

1. **Find the inverse of the matrix** \( M = \begin{bmatrix} 2 & 3 & 4 \\ 3 & 5 & 7 \\ 1 & 2 & 4 \end{bmatrix} \) using elementary row operations. Step 1: Write the augmented matri

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. This is a 2x2 matrix with elements \(a, b, c, d\) arranged as:

1. The problem is to understand what a nilpotent matrix is and what is meant by the period of a nilpotent matrix. 2. A nilpotent matrix $N$ is a square matrix such that for some po

1. **State the problem:** We are given a matrix $$A = \begin{bmatrix} 5 & -2 & -1 \\ 5 & -1 & 1 \\ 7 & 4 & \lambda \end{bmatrix}$$ and asked to find the characteristic polynomial b