📘 linear algebra

1. **Problem statement:** Given linear transformations $f: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $f(x,y) = (2x + 5y, 3x + 8y)$ and $g(x,y) = (x + 2y, 4x + 5y)$, find: i) The ma

1. The problem asks whether the determinant of an $n \times n$ matrix $A$ in echelon form is the product of the elements along its main diagonal. 2. Recall that the determinant of

1. **State the problem:** Find the determinant of matrix $$A=\begin{bmatrix} 1 & 3 & -2 \\ 3 & 0 & 1 \\ -1 & 2 & -4 \end{bmatrix}$$ using the basket method. 2. **Recall the basket

1. The problem asks to find the product of matrices $A$ and $B$, denoted as $AB$. 2. To multiply two matrices, the number of columns in matrix $A$ must equal the number of rows in

1. The statement "Using Cramer's Rule involves computing determinants for both the coefficient matrix and matrices formed by replacing columns with constant vectors" is TRUE. Expla

1. The first statement is: "Cramer's Rule can only be applied to systems with exactly as many equations as unknowns." This is TRUE. Cramer's Rule requires a square coefficient matr

1. **Stating the problem:** We need to find the product of the column vector $$\begin{bmatrix}0 \\ -2\end{bmatrix}$$ and the row vector $$\begin{bmatrix}-1 & 0 & -1\end{bmatrix}$$.

1. **State the problem:** Multiply the 2x2 matrix $$\begin{bmatrix}1 & 0 \\ 3 & -1\end{bmatrix}$$ by the 2x1 vector $$\begin{bmatrix}2 \\ 1\end{bmatrix}$$. 2. **Recall the matrix m

1. **State the problem:** Find the value of $z$ in the system of linear equations given by the matrix equation:

1. **State the problem:** Solve the system of equations using row operations to transform the augmented matrix \((A|b)\) into echelon or reduced echelon form. 2. **Write the system

1. **Stating the problem:** We are given a 4x4 matrix \( A \) and asked to find \( z \) using Cramer's rule. The matrix \( A \) is:

1. **Problem Statement:** Given vectors $v_1 = \begin{bmatrix}1 \\ 1 \\ 0\end{bmatrix}$, $v_2 = \begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}$, and $v_3 = \begin{bmatrix}0 \\ 1 \\ 1\end{

1. **State the problem:** We are given two vectors:

1. **Problem statement:** Determine for which values of $k$ the matrix $$

1. **State the problem:** Rewrite the system of differential equations in matrix notation with clear rows and columns. 2. **Matrix notation:** The system can be written as $$\frac{

1. **Problem statement:** (a) Find the product of matrices $A$ and $B$ where

1. **Stating the problem:** We have a triangle with vertices $P(-3,2)$, $Q(0,-1)$, and $R(2,-1)$. A transformation matrix $M$ maps these points to $P^\prime(-7,2)$, $Q^\prime(2,-1)

1. Diberikan matriks A sebagai berikut: $$A=\begin{bmatrix}3 & 5 & -1 \\ 2 & 3 & 1 \\ -1 & 2 & 2\end{bmatrix}$$

1. **Problem:** Find the Adjoint Matrix of the given matrix $$D = \begin{bmatrix} 11 & 9 & -4 \\ -2 & 0 & 1 \\ 0 & -6 & 1 \end{bmatrix}$$. 2. **Formula and Explanation:**

1. **Problem statement:** Find scalars $a$ and $b$ such that $$a\mathbf{u} + b\mathbf{v} = (1, -4, 9, 18)$$ where $$\mathbf{u} = (1, -1, 3, 5)$$ and $$\mathbf{v} = (2, 1, 0, -3).$$

1. **State the problem:** We need to solve the matrix equation