📘 Linear Algebra

1. Problem: You asked 'On for matrix step by step'. Please provide the matrix entries and specify the operation you want (for example determinant, inverse, row reduction, or eigenv

1. Problem statement: (a) Given a matrix $$A=\begin{bmatrix} a & 1 & 1 & 1 \\ 1 & a & 1 & 1 \\ 1 & 1 & a & 1 \\ 1 & 1 & 1 & a \end{bmatrix}$$

1. **Menentukan determinan Matriks 2x2 dan 3x3** Untuk matriks 2x2 $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, determinan dihitung dengan rumus:

1. **Problem statement:** We are given the matrix $$A = \begin{bmatrix}\mathcal{a} & 1 & 1 & 1 \\ 1 & \mathcal{a} & 1 & 1 \\ 1 & 1 & \mathcal{a} & 1 \\ 1 & 1 & 1 & \mathcal{a}\end{

1. a) State the Second Fundamental Theorem of Subspace: If $W$ is a subspace of $V$, then $W$ must be closed under vector addition and scalar multiplication, and contain the zero v

1. The term "P matrix" can refer to several different matrices depending on context, so let's clarify the most common meanings. 2. In linear algebra, a "P matrix" sometimes refers

1. The problem is to understand and apply Cramer's Rule to solve a system of linear equations. 2. Cramer's Rule states that for a system of $n$ linear equations with $n$ unknowns,

1. **State the problem:** We need to find matrix transformations for rotating an image by a positive $\frac{\pi}{2}$ turn (90 degrees) about the origin, scaling by 2 times, reflect

1. **نوضح المشكلة:** نريد إيجاد القيم الذاتية ($\lambda$) والاشعة الذاتية (المتجهات الذاتية) لمصفوفة معينة $A$. 2. **خطوة أولى:** لحساب القيم الذاتية لمصفوفة $A$، نحل المعادلة التا

1. **Stating the problem:** Find the eigenvalues and corresponding eigenvectors of the matrix $$A=\begin{pmatrix}8 & -6 & 2 \\ -6 & 7 & -4 \\ 2 & -4 & 3 \end{pmatrix}$$

1. **Problem statement:** Find the eigenvalues and eigenvectors of the matrix $$A = \begin{pmatrix} 11 & 4 & 7 \\ 7 & 2 & 5 \\ 10 & 4 & 6 \end{pmatrix}.$$

1. Problemi: Identifikoni nëse dy drejta janë të njëjta. Drejtat jepen:

1. সমস্যা জানা: দিয়ে হয়েছে $A^{-1} = \begin{vmatrix} \frac{5}{7} & \frac{1}{7} \\ \frac{3}{7} & \frac{2}{7} \end{vmatrix}$. আমাদের $A^2 + 2A$ এর মান বের করতে হবে। প্রথমে $A$ নির্

1. The user asks to "solve this one matrix" but no matrix is provided in the prompt. 2. To assist, please provide the matrix or the related problem statement, such as solving a sys

1. **Problem Q11:** Compute the eigenspaces of the matrix $$\begin{bmatrix} 1 & 0 & -2 & 2 \\ 1 & 1 & 2 & 1 \end{bmatrix}$$ 2. Since this matrix is 2x4 (not square), it does not ha

1. The problem is to solve a system of linear equations using Gaussian elimination. 2. Write the system as an augmented matrix.

1. **State the problem:** Find the eigenvalues and corresponding eigenvectors for the matrix $$A=\begin{bmatrix}-2 & 2 & -1 \\ -2 & -3 & -6\end{bmatrix}$$

1. **Find the rank of the matrix** Given matrix

1. **State the problem:** Find the eigenvalues of the matrix $$\begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix}$$. 2. **Recall the definition:** The eigenvalues $$\lambda$$ satisfy the c

1. Problem 1: Use Cayley-Hamilton theorem to find $$A^8 - 5A^7 + 7A^6 - 3A^5 + 4A^4 - 5A^3 + 8A^2 - 2A + I$$ where $$A=\begin{bmatrix} 2 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 2 \end{bmat

1. **Stating the problem:** We have a system of linear equations: $$