📘 linear algebra

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.

1. **State the problem:** Given matrices representing points per outcome and match results for two teams, we want to find the product matrix $RP$. 2. **Define the points matrix $P$

1. The problem asks which of the given matrices are upper triangular. 2. Recall that an upper triangular matrix is a square matrix where all elements below the main diagonal are ze

1. The problem asks about the condition for matrix multiplication $A \times B$ to be defined. 2. Let matrix $A$ have dimensions $m \times n$ (rows by columns) and matrix $B$ have d

1. Stating the problem: Calculate the determinant of matrix D given the elements 3 and 4 arranged as a 2x1 matrix. 2. Clarification: The determinant is defined for square matrices

1. **State the problem:** Find the determinant $D$ of the $3\times3$ matrix: $$D = \begin{bmatrix} 1 & 2 & 1 \\ 3 & 1 & 2 \\ 2 & 1 & 4 \end{bmatrix}$$

1. The problem asks us to identify which of the given options is NOT a square matrix. 2. A square matrix is defined as a matrix with the same number of rows and columns.

1. The problem is to analyze the matrix $$\begin{bmatrix}a&b\\c&d\end{bmatrix}$$. 2. This is a general 2x2 matrix with elements $a, b, c, d$.

1. Problem i: Find $r$ and $s$ such that $AB^T = 0$, where $A = [1, r, 1]$, $B = [-2, 2, s]$ Calculate $$AB^T = 1 \times (-2) + r \times 2 + 1 \times s = -2 + 2r + s = 0.$$ Rearran

1. Given the matrix $$B = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix},$$ we need to find its inverse $$B^{-1}$$. 2. Notice that $$B$$ is an upper triangular m

1. **Stating the problem:** We are given an equation involving determinants of two $3 \times 3$ matrices:

1. The problem is to understand the properties of the given 3x3 matrix: $$\begin{bmatrix} 1 + a^2 - b^2 & 2ab & -2b \\ 2ab & 1 - a^2 + b^2 & 2a \\ 2b & -2a & 1 - a^2 - b^2 \end{bma

1. We need to find the determinant of the 2x2 matrix \(\begin{pmatrix} K & K \\ 4 & 2K \end{pmatrix}\) using cofactor expansion along the first column. 2. The determinant is comput

1. **State the problem:** We are given the matrix \( \begin{bmatrix}

1. **Stating the problem:** Determine if the matrix

1. The problem involves solving the system of linear equations represented by the augmented matrix: $$\begin{bmatrix}1 & -1 & 2 & 4 & 6 & | & 2 \\ 0 & 1 & 2 & 1 & -1 & | & -1 \\ 0

1. **State the problem:** We are given matrix $$A=\begin{bmatrix}1 & 5 & 3 \\ -1 & -4 & -1 \\ -2 & -7 & 0\end{bmatrix}$$ and need to find the first row of its row-reduced echelon f

1. The problem is to determine the solution set of the linear system represented by the augmented matrix: $$\begin{bmatrix} 1 & 1 & 1 & | & 5 \\ 1 & -1 & 5 & | & 3 \\ -2 & 2 & -10

1. We are given the matrix $$\begin{bmatrix} 1 & 0 & a+b \\ a & a & 0 \\ 0 & 0 & b \end{bmatrix}$$