#### Matrix - Linear System Equations

A magic square is a square table of natural numbers, side $n$, where the sum of row numbers, column numbers, and diagonal numbers is constant, none of which repeats. Enter the values of $a$, $b$ and $c$ so that $A=\begin{bmatrix} a&2&9\\8&b&4\\3&10&c \end{bmatrix}$ be a matrix representing a magic square.

# System of linear equations

Modeling
13

#### Power of a matrix 2

Let $A=\begin{bmatrix} -0.5& -1 & 0.5\\1 &0.5 & -1\\0.5 &0 &0.5\end{bmatrix}$, $M=A^2$ and $N=A^3$. The entries $m_{32}$ and $n_{23}$ of $M$ and $N$ are (respectively) equal to:

# Operation

Reasoning
13

#### Matrix operations 1

Consider identity matrix $I_3$ and the matrices $$A=\begin{bmatrix} 1&-1&2\\-0.5&0&1 \end{bmatrix},\quad B=\begin{bmatrix} 1&1&-1\\2& 0& 0\\-1&0&1 \end{bmatrix}$$ $$C=\begin{bmatrix} 1&0\\2& -1\\-1&1 \end{bmatrix}$$ Check which of the following operations are possible: $(a1)\ C-2A^T\quad (a2)\ 3A-C\quad (a3)\ (I_3+2B)^T\quad$

# Operation

Training
13

#### Operations mix

Consider the real matrices $A=[a_{ij}],\ i,j=1,2,3:\ \ a_{ij}=\left\{ \begin{array}{l} i-2, \quad i>j\\ 0, \quad \quad i=j\\ ij+1, \quad i<j \end{array} \right.$, $B=\begin{bmatrix} \frac{1}{3}&\frac{2}{3}&7\\\frac{1}{3}&-\frac{1}{3}&1\\3&3&\frac{1}{3}\end{bmatrix}$, $C=\begin{bmatrix} 1&2\\0&1\\-1&0\end{bmatrix}$ and $D=\begin{bmatrix} 1&2&3\\-1&0&1\end{bmatrix}$. Compute $M=A^2 - 3B + (CD)^T$.

# Operation

Learning
13

#### Linear system equations and Gauss elimination method

Applying the Gauss elimination's method, the solution of the linear system equations $$\left\{ \begin{array}{l} 2x+y-z=1\\ -x+2y-3z=1\\ 4x+z=2 \end{array} \right.$$ is:
# Matrices
# System of linear equations

Training
13

#### Matrix representation of a system of linear equations

Consider the system of linear equations $$\left\{ \begin{array}{l} x-y+2z=3\\ -3y-z=-5\\ 3x-y+4z=7 \end{array} \right.$$ The second and the fourth columns of the matrix that represents the system are:

# System of linear equations

Training
13