📘 linear algebra

1. The problem asks to find the determinant of the matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$. 2. The determinant of a 2x2 matrix $$\begin{bmatrix}a & b \\ c & d\end{bma

1. The problem is to multiply the row vector $\begin{bmatrix} 2 & 3 \end{bmatrix}$ by the matrix $\begin{bmatrix} 2 & -1 & -3 \\ 0 & 4 & 5 \end{bmatrix}$.\n\n2. Matrix multiplicati

1. Stated problem: Multiply the 1x2 row vector $\begin{bmatrix} 2 & 3 \end{bmatrix}$ by the 2x3 matrix $\begin{bmatrix} 2 & -1 & -3 \\ 0 & 4 & 5 \end{bmatrix}$.\n\n2. Matrix multip

1. The problem asks to find the product of a 1x2 row vector and a 2x2 matrix. 2. The row vector is given as $\begin{bmatrix} 2 & 1 \end{bmatrix}$.

1. **State the problem:** We are given matrices

1. المشكلة: لدينا متجه عمودي $\begin{bmatrix}3 \\ 6\end{bmatrix}$ ونريد فهم العملية التي تحول هذا المتجه إلى العدد 3، ثم هناك عملية طرح بين متجه $\begin{bmatrix}3 \\ 6\end{bmatrix}

1. **State the problem:** We have a triangle with vertices A(1,2), B(3,2), C(3,1) in the plane. We want to apply the composite linear transformation \(T_2 \circ T_3 \circ T_1\) to

1. **State the problem:** Solve the system $Ax = b$ where $$A = \begin{bmatrix} 2 & 1 & -1 \\ 4 & 5 & 0 \\ -2 & 3 & 8 \end{bmatrix}, \quad b = \begin{bmatrix} 3 \\ 7 \\ 1 \end{bmat

1. **State the problem:** We have matrix $$A = \begin{bmatrix} 2 & 6 & 23 \\ -5 & -12 & -45 \\ 5 & 15 & 60 \end{bmatrix}$$ and the system:

1. **State the problem:** Given matrix $$A = \begin{bmatrix} -2 & -2 & 1 \\ -2 & -1 & -2 \\ 3 & -2 & 0 \end{bmatrix}$$, find: (a) The determinant of $$A$$.

1. **State the problem:** Find elementary matrices $E_1, E_2, E_3, E_4$ such that $$E_4 E_3 E_2 E_1 \begin{pmatrix} 1 & 0 & 0 \\ 7 & 2 & 0 \\ 5 & 8 & 3 \end{pmatrix} = \begin{pmatr

1. **Problem Statement:** Find the value(s) of $h$ for which the vectors in each set are linearly dependent. Vectors are linearly dependent if the determinant of the matrix formed

1. **Problem:** Determine if the vectors \(\begin{bmatrix}0 \\ 0 \\ 2\end{bmatrix}\), \(\begin{bmatrix}0 \\ 5 \\ -8\end{bmatrix}\), and \(\begin{bmatrix}-3 \\ 4 \\ 1\end{bmatrix}\)

1. **State the problem:** We are given a linear transformation defined by the matrix $$A = \begin{bmatrix} 1 & 1 & -2 \\ 2 & -1 & 1 \\ 3 & 1 & -1 \end{bmatrix}$$

1. **State the problem:** We want to express the vector $\mathbf{v} = (5, 5, 5, 5)$ as a linear combination of the vectors $\mathbf{v_1} = (1, 2, 3, 4)$, $\mathbf{v_2} = (2, 3, 4,

1. **Stating the problem:** Solve the system of linear equations (SPL) using row operations (OBE) on the augmented matrix:

1. **Nêu bài toán:** Cho ánh xạ $f : \mathbb{R}^3 \to \mathbb{R}^3$ xác định bởi $$f(x_1, x_2, x_3) = (x_1 + x_2, x_1 + 2x_2 + x_3, 3x_1 - 2x_3).$$

1. **Stating the problem:** We are given matrices \(F\), \(A\), and \(B\) as follows: \[

1. Problem 16: Given matrices $$A = \begin{pmatrix}2 & -2 \\ -1 & 3\end{pmatrix}, \quad AB = \begin{pmatrix}4 & -2 \\ 0 & 7\end{pmatrix}$$

1. Problem 16: Find matrix \( B \) given \( A \) and \( AB \). Step 1: Write down the matrices.

1. The problem states to find the determinant value of a singular matrix. 2. A singular matrix is defined as a square matrix that does not have an inverse.