📘 Linear Algebra

1. **Problem 10:** Determine if each matrix is Hermitian, Skew-Hermitian, or neither. 2. Recall definitions:

1. **State the problem:** We are given a matrix $$A = \begin{pmatrix} 2 & -1 \\ 3 & -2 \end{pmatrix}$$ and need to find the sum $$S = I + A + A^2 + A^3 + \cdots + A^{11} + A^{12}$$

1. **State the problem:** We want to calculate the regression coefficient vector $\beta$ using the formula: $$\beta = (X^T X)^{-1} X^T Y$$

1. نبدأ بتحديد المشكلة: لدينا نقاط في الفضاء ثلاثي الأبعاد أ، ب، ج، د، ع، ونريد إيجاد انعكاس كل نقطة بالنسبة للخط الذي يحقق المعادلة $x = y = z$. 2. الخط $x = y = z$ هو خط يمر عبر

1. The problem states that $\det(AT) \times \det(A) = 5$, where $AT$ is the transpose of matrix $A$. 2. Recall the property of determinants: $\det(AT) = \det(A)$ for any square mat

1. **Problem Statement:** (a) Define what it means for a vector space $V$ to be the direct sum of two subspaces $W_1$ and $W_2$, denoted as $V = W_1 \oplus W_2$.

1. **Problem 5(a): Show that $M_{2\times 2}(\mathbb{R})$ forms a vector space under standard matrix addition and scalar multiplication.** - The set $M_{2\times 2}(\mathbb{R})$ cons

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

1. **State the problem:** We have the system of equations: $$\begin{cases} x + y + z = 5 \\ x + 2y + z = 9 \\ x + y + (a^2 - 5)z = a \end{cases}$$

1. **State the problem:** Find the inverse of the matrix $$\begin{bmatrix} 9 & 1017 & 954 \\ 1017 & 115571 & 7690 \\ 954 & 107690 & 101772 \end{bmatrix}$$

1. **State the problem:** Solve the system of linear equations using Gaussian elimination: $$\begin{cases} 3x + 5y + 7z = 1 \\ 4x + 2y + 4z = 1 \\ x + 9y + 8z = 1 \\ 8x + 8y + 2z =

1. Problem 7: Given matrices $$A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}, B = \begin{bmatrix}2 & 3 \\ 0 & 1\end{bmatrix}, C = \begin{bmatrix}5 & 2 \\ -1 & 0\end{bmatrix}$$

1. The problem is to analyze the vector $A = (2, -3, 0, 4, 2, -2)$. 2. We can find the magnitude (length) of the vector $A$ using the formula:

1. **State the problem:** We are given the vector equation:

1. **State the problem:** We need to find the vector \( \begin{bmatrix} x \\ y \end{bmatrix} \) given by the equation $$

1. **State the problem:** We have a 3x3 matrix $A$ composed of $\cos\theta$ and $\sin\theta$ terms, and we want to show that $A$ is invertible and find its inverse $A^{-1}$. 2. **A

1. **State the problem:** We are given two matrices and asked to find the period of the first matrix and determine which of the two given matrices is idempotent or involutory. 2. *

1. **State the problem:** Find the index of nilpotency for each given 3x3 matrix. The index of nilpotency is the smallest positive integer $k$ such that $A^k = 0$. If no such $k$ e

1. The problem asks to find a unit vector \( \mathbf{u} \) in the same direction as the vector \( \mathbf{v} = \begin{pmatrix}4 \\ -2 \\ 1\end{pmatrix} \). 2. A unit vector has len

1. **State the problem:** We are given a vector $\mathbf{v} = \begin{pmatrix}4 \\ -2 \\ 1\end{pmatrix}$ and need to find a unit vector $\mathbf{u}$ in the same direction as $\mathb

1. **State the problem:** Find the inverse of the matrix $$A=\begin{bmatrix}1 & -1 & 2 \\ 0 & 2 & -1 \\ 3 & 1 & 0\end{bmatrix}$$. 2. **Calculate the determinant of A:**