📘 linear algebra

1. **Matrix multiplication problem:** Given matrices $$

1. **Problem statement:** Given a matrix $A = [a_{ij}]$ of size $3 \times 4$, show that $I_3 A = A$ where $I_3$ is the $3 \times 3$ identity matrix. 2. **Recall the identity matrix

1. The problem is to find the 3x3 identity matrix and understand its meaning. 2. The identity matrix, often denoted as $I_n$ for an $n \times n$ matrix, is a square matrix with 1's

1. State the problem: Find the determinant of the matrix $$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$. 2. Formula: For a 2x2 matrix $$\begin{bmatrix} a & b \\ c & d \end{bm

1. **Problem Statement:** Determine whether the set of all $2 \times 2$ matrices of the form $$\begin{bmatrix} a & a + b \\ a + b & b \end{bmatrix} \quad a,b \in \mathbb{R}$$

1. The problem asks to write the zero vector in the space of $3 \times 4$ matrices over the field $F$. 2. The zero vector in a matrix space is the matrix where every entry is zero.

1. **Problem statement:** Given matrices \(A = \begin{pmatrix} 2 & -3 \\ \end{pmatrix}\) and \(B = \begin{pmatrix} 1 & 0 \\ -3 & \end{pmatrix}\), calculate \(AB\). 2. **Matrix mult

1. The problem appears to be incomplete or unclear, but it seems to involve an expression with an exponent, possibly $e^{LU}$. 2. If the problem is to evaluate or simplify $e^{LU}$

1. Тодорхойлогчийн (детерминантын) утга нь квадрат матрицын мөрүүдийг хооронд нь сольж, үржүүлж, нэмэх зэрэг үйлдлүүдэд хэрхэн өөрчлөгдөхийг харуулдаг. 2. Хэрэв матрицын нэг мөрийг

1. **Stating the problem:** We need to determine which of the given 3x3 matrices are in row-echelon form. 2. **Definition and rules of row-echelon form:**

1. **State the problem:** Find the matrices $2A + AB^T$ and $AB$ given

1. **State the problem:** Find the determinant and inverse of matrix $$A=\begin{bmatrix}5 & -7 & 1 \\ 6 & -8 & -2 \\ 3 & -1 & 6\end{bmatrix}$$. 2. **Find the determinant of matrix

1. **Problem Statement:** Test the consistency of the system and solve if consistent: $$\begin{cases} x + y + z = 3 \\ x + 2y + 3z = 4 \\ x + 4y + 9z = 6 \end{cases}$$

1. **Problem statement:** Given matrices $$A=\begin{pmatrix}-3 & 0 & 5 \\ -2 & 1 & -1 \\ -1 & 4 & 0\end{pmatrix},\quad B=\begin{pmatrix}5 & -1 & 2 \\ 3 & 2 & 1 \\ -1 & 4 & 1\end{pm

1. The problem asks about the determinant of a matrix $M$ when $M$ is not a square matrix. 2. The determinant, denoted as $\det(M)$, is defined only for square matrices, which mean

1. **State the problem:** We are given a 5x5 matrix $$

1. **Problem:** Determine the rank of the matrix $$\begin{bmatrix} 91 & 92 & 93 & 94 & 95 \\ 92 & 93 & 94 & 95 & 96 \\ 93 & 94 & 95 & 96 & 97 \\ 94 & 95 & 96 & 97 & 98 \\ 95 & 96 &

1. The problem asks to change questions 8, 9, 10, 15, 16, 17, 23, and 24 by matrices. 2. Since the user did not provide specific questions, I will explain how to represent typical

1. **Solve the system using Gauss Elimination Method:** Given system:

1. **State the problem:** Solve the system of equations using Gauss Elimination Method: $$\begin{cases} 4x - 5y + 6z = 12 \\ 7x + 3y - 2z = -5 \\ 5x - 4y + 8z = 10 \end{cases}$$

1. **State the problem:** Solve the system of linear equations using the Jacobi Method: $$\begin{cases} 5x - y + z = 10 \\ 2x + 8y - z = 12 \\ x - y + 4z = 6 \end{cases}$$