📘 linear algebra

1. **State the problem:** We are asked to add two matrices \(B\) and \(C\) where \(B = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}\) and \(C = \begin{bmatrix} 7 & -1 & 5 \\ 0 & -3

1. **State the problem:** Given vectors $a = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$ and $b = \begin{pmatrix} 5 \\ -1 \end{pmatrix}$, find the column vectors for: (i) $a + b$

1. **State the problem:** Solve the system of linear equations using Gaussian elimination: $$\begin{cases}-2x - 3y + 2z + u = 0 \\ -y + z + y = 1 \\ -4x + 2y - u = 0 \end{cases}$$

1. **State the problem:** Multiply the matrices $$A = \begin{bmatrix}-2 & -4 & 0 \\ 0 & 4 & -1\end{bmatrix}$$

1. The problem is to understand vector addition both algebraically and graphically. 2. Given vectors:

1. **Problem Statement:** Solve the system of linear equations using the inverse matrix method: $$\begin{cases} 3x + 2y - z = 5 \\ 2x - 2y + 4z = -2 \\ -x + 0y + 5z = 17 \end{cases

1. **Problem Statement:** We are given a 3x3 scale matrix $A$ with distinct non-zero diagonal elements $a_{11}$, $a_{22}$, and $a_{33}$. We need to calculate the determinant of thi

1. **State the problem:** Find the inverse of matrix $$A = \begin{pmatrix} 2 & 4 & 2 \\ -3 & 4 & 3 \\ -1 & 1 & 3 \end{pmatrix}$$ using the Gauss-Jordan elimination method. 2. **Met

1. **Problem statement:** Calculate the determinant of matrix $A = (a_{ij})_{4 \times 4}$ where $$a_{ij} = \begin{cases} \log_2(32 - i), & i < j \\ \log_3\left(\frac{27}{81}\right)

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 in two rows and two columns

1. **Problem:** Show that the linear operator $T: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $$u_1 = 2x_1 + x_2, \quad u_2 = 3x_1 + 4x_2$$

1. **Problem:** Find the value of $p$ such that the determinant of the matrix $$\begin{pmatrix}-2 & 1 & 1 \\ p & 1 & 3 \\ 1 & 2 & -1 \end{pmatrix} = 22$$

1. **Stating the problem:** We are given a matrix \(H\) and an expression involving vectors and matrices, as well as a calculus expression for \(a\).

1. **State the problem:** We are given a system of linear equations representing a Markov chain's stationary distribution $\pi = (\pi_1, \pi_2, \pi_3)$ with transition probabilitie

1. **Stating the problem:** We are given a system of linear equations involving variables $\pi_1$, $\pi_2$, and $\pi_3$:

1. **Problem 1: Show that** $A + (B + C) = (A + B) + C$ for given matrices. 2. **Given:**

1. **Problem Statement:** Verify the transpose properties for matrices \(A\) and \(B\) given as: $$A = \begin{pmatrix} 1 & -1 & 2 \\ 5 & -4 & 3 \\ 1 & -2 & -3 \end{pmatrix}, \quad

1. **State the problem:** We need to show that the vectors $\mathbf{v_1} = (3,0,-3)$, $\mathbf{v_2} = (-1,1,2)$, $\mathbf{v_3} = (4,2,-2)$, and $\mathbf{v_4} = (2,1,1)$ are linearl

1. **Problem Statement:** Solve the system of linear equations: $$\begin{cases} 3x + 2y + z = 2 \\ 4x + 2y + 2z = 8 \\ x - y + z = 4 \end{cases}$$

1. **State the problem:** We are given matrices $$A = \begin{bmatrix}1 & 2 & 3 \\ -1 & 1 & 2 \\ 1 & 2 & 4\end{bmatrix}, \quad B = \begin{bmatrix}-3 \\ -6 \\ -9\end{bmatrix}$$

1. **Problem statement:** Given the matrix $$M = \begin{pmatrix} 0 & 0 & a \\ b & 0 & 0 \\ 0 & c & 0 \end{pmatrix}$$