📘 Linear Algebra

1. **Problem:** Use the Subspace Test to determine which of the sets are subspaces of $\mathbb{R}^3$. a. All vectors of the form $(a, 0, 0)$.

1. **Problem Statement:** Imagine a company wants to analyze the performance of two products over time. They collect data on sales and customer satisfaction, represented by a matri

1. **State the problem:** We need to reduce the matrix $$\begin{bmatrix} 1 & 3 & 5 \\ -6 & 8 & 3 \\ -4 & 6 & 5 \end{bmatrix}$$

1. **State the problem:** Find the eigenvectors of a given matrix $A$. 2. **Recall the definition:** Eigenvectors $\mathbf{v}$ satisfy the equation $$A\mathbf{v} = \lambda \mathbf{

1. **State the problem:** Find the eigenvalues and corresponding eigenvectors of the matrix $$A = \begin{pmatrix}-2 & 2 & -3 \\ 2 & 1 & -6 \\ -1 & -2 & 0\end{pmatrix}$$

1. **Problem Statement:** You are solving a least squares problem involving matrices $A$, $b$, and the normal equations $A^T A x = A^T b$ to find the vector $x = (x_1, x_2, x_3)$.

1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\) and its properties. 2. This is a 2x2 matrix with elements \(a, b, c, d\).

1. **Problem Statement:** Find the least squares solution of the linear system given by $$\begin{cases} 3x_1 + x_2 - x_3 = -3 \\ x_1 + 2x_2 = -3 \\ x_2 + 2x_3 = 8 \end{cases}$$

1. **Problem statement:** Solve the system of equations using the Gauss-Seidel iteration method: $$\begin{cases} 20x + y - 2z = 17 \\ 3x + 20y - z = -18 \\ 2x - 3y + 20z = 25 \end{

1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix} 1 & 0 & 1 \\ -1 & 2 & 2 \\ 1 & 1 & 2 \end{bmatrix}$$ using the adjoint method. 2. **Formula and Impor

1. **State the problem:** Solve the system of linear equations using the Gauss-Seidel iteration method: $$\begin{cases} 2x + y + z = 4 \\ x + 2y + z = 4 \\ x + y + 2z = 4 \end{case

1. **Problem:** Given matrices $$A=\begin{pmatrix}-2 & 3 & 3 \\ 1 & 0 & -4 \\ -3 & 1 & -1\end{pmatrix}, \quad B=\begin{pmatrix}1 & -3 & 0 \\ -1 & 2 & 2 \\ 6 & -1 & 5\end{pmatrix}$$

1. State the problem: Write 20 standard multiple choice questions on determinant, minor, cofactor, inverse, equality of matrix, symmetry and skew symmetric, operation on matrix. 2.

1. **Problem:** What is the determinant of a 2x2 matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$? 2. **Formula:** The determinant of $A$ is given by $$\det(A) = ad - bc$$

1. **Problem:** Prove or disprove that $M_{2\times 2}(\mathbb{R}) = \{\begin{bmatrix}a & b \\ c & d\end{bmatrix} \mid a,b,c,d \in \mathbb{R}\}$ is a vector space over $\mathbb{C}$.

1. **Problem Statement:** Given matrices $$A=\begin{pmatrix}2 & -1 \\ 0 & -4 \\ -5 & 3\end{pmatrix}, B=\begin{pmatrix}5 & -2 & 6 \\ -1 & 4 & -2\end{pmatrix}, C=\begin{pmatrix}2 & 1

1. **Stating the problem:** We want to multiply two matrices, say matrix $A$ of size $m \times n$ and matrix $B$ of size $n \times p$. The result will be a matrix $C$ of size $m \t

1. **State the problem:** We need to show that the matrix $$A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}$$ is idempotent. This means we want to verify

1. **Problem Statement:** Prove that if $A$ is a non-singular matrix of order $n$, then $\operatorname{adj}(A) = \det(A) A^{-1}$.

1. **State the problem:** Find matrix $K$ such that:

1. The problem is to multiply the scalar 5 by the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. The formula for scalar multiplication of a matrix is: