📘 linear algebra

1. **Find the inverse of matrix** $$A=\begin{pmatrix}2 & 1 & 3 \\ 1 & 0 & 2 \\ 4 & 1 & 8\end{pmatrix}$$ 2. Calculate the determinant $$\det(A) = 2(0\cdot8 - 2\cdot1) - 1(1\cdot8 -

1. **State the problem:** We are given two 3x3 matrices A and B: $$A = \begin{pmatrix}7 & 8 & 6 \\ 1 & 3 & 9 \\ -4 & 3 & -1\end{pmatrix}, \quad B = \begin{pmatrix}4 & 11 & -3 \\ -1

1. The problem shows two 3x3 matrices labeled (c) and (d). 2. Matrix (c) is:

1. **State the problem**: Given matrix

1. The problem asks to determine which of the given matrices are in row echelon form (REF) and which are in reduced row echelon form (RREF). 2. Matrix (g):

1. The statement is det(A - B) = det(A) - det(B). We need to determine if this is true or false. 2. Recall the property of determinants: $$\det(A + B) \neq \det(A) + \det(B)$$ in g

1. First, evaluate $\det(2A^T)$ given $\det(A) = 3$.\n Since $A$ is a $3 \times 3$ matrix, scaling by 2 multiplies the determinant by $2^3 = 8$. Also, the determinant of the transp

1. **Problem statement:** For the matrix $$A = \begin{pmatrix} 3 & -2 & 2 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{pmatrix},$$ verify which of the vectors $$\mathbf{v}_a = \begin{pmatrix} 3

1. Өгөгдсөн матрицуудын урвуу матрицыг ол. Матрицыг урвагчаар үржүүлж, тэнцэтгэлт матриц үүсгэх замаар олно. 2. Матрицын урвууг олох үндсэн арга бол эгнээгээр төлөөлж, эрэмбэлж, эл

1. **Problem statement:** Given matrix $$ M = \begin{bmatrix} 3 & -6 & 2 \\ 6 & 2 & -3 \\ 2 & 3 & 6 \end{bmatrix} $$

1. **Статья**: Матрицуудыг нэмэх, тоогоор үржүүлэх, матрицаар үржүүлэх, хөрвүүлэх гэх мэт үйлдлүүдийг биелүүлэх. ---

1. The problem is to analyze the three given 2x2 matrices which contain numeric and symbolic entries including fractions and negative signs. 2. The first matrix is:

1. **بيان المشكلة:**\nنحن بصدد التحقق من نواتج حاصل ضرب بعض المصفوفات، ونتأكد مما إذا كانت بعض الأزواج من المصفوفات تمثل "تغيرًا ضربياً" أي أن إحداها يمكن أن تُنتج بواسطة ضرب الأخر

1. The given matrix is \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \). 2. In matrix algebra, such a matrix is a 2x2 matrix with elements \(a, b, c, d\).

1. **نص المشكلة:** لدينا مصفوفة $P = X - X'$

1. The problem states: Let $T: W \to V$ be a linear transformation where $W$ and $V$ are vector spaces. If $W$ is finite dimensional, prove that the nullity of $T$ plus the rank of

1. **State the problem:** We are given the system of linear equations: $$x + y + z = 6$$

1. Solve the system of equations using Gauss Elimination method: Given equations:

1. The problem is to find the rank of a matrix. 2. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix.

1. **State the problem:** We need to show that $$A^3 - A^2 + A - 2B = 0$$ where $$A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}$$ and $$B = \begin{bmatrix}

1. **Problem statement:** We are given four statements about determinants of $n \times n$ matrices $A$ and $B$. We need to check which statements are true. 2. **Statement A:** $\de