📘 linear algebra

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and Rules:** The inverse of a matrix

1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix}$$. 2. **Formula for Inverse of a 2x2 Matrix:** For a matrix $$A = \begin

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **Problem Statement:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and Rules:** The inverse of a matrix

1. **State the problem:** We need to find the rank of the matrix $$\begin{bmatrix}3 & 2 & 1 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}$$. 2. **Recall the definition:** The rank of a ma

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **State the problem:** Find the inverse of the matrix $$A = \begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 6 & 7 & 8\end{bmatrix}$$. 2. **Formula and rules:** The inverse of a matrix

1. **State the problem:** Find the eigenvalues of the matrix $$\begin{bmatrix}4 & 1 \\ 2 & 3\end{bmatrix}$$. 2. **Formula:** Eigenvalues $\lambda$ satisfy the characteristic equati

1. **State the problem:** We are given vectors $v_1 = (1,2,1)$, $v_2 = (2,9,0)$, and $v_3 = (3,3,4)$ which form a basis $S$ for $\mathbb{R}^3$. We want to find the coordinate vecto

1. The problem is to calculate the adjugate (or adjoint) matrix of a given square matrix. 2. The adjugate matrix is the transpose of the cofactor matrix.

1. **Stating the problem:** We need to find the unknown $x$ from the equation involving determinants:

1. **Stating the problem:** Prove the equivalence of the statements in Theorem 1.6.4 about an $n \times n$ matrix $A$: (a) $A$ is invertible.

1. Bài toán yêu cầu xác định giá trị của $m$ sao cho ảnh của ánh xạ tuyến tính $f : P_2 \to P_2$ có chiều bằng 2. 2. Ánh xạ $f$ được định nghĩa trên đa thức bậc 2:

1. **State the problem:** Find the rank of the matrix $$\begin{bmatrix}1 & 3 & 4 \\ 2 & 1 & 2 \\ 3 & 1 & 1\end{bmatrix}$$ using elementary row transformations. 2. **Recall:** The r

1. **State the problem:** Find the rank of the matrix using elementary transformations. Given matrix:

1. Problem: Verify the associative law for matrix addition for matrices A, B, and C. 2. The associative law for matrix addition states that for any matrices A, B, and C of the same