#### Matrix operations 3

Consider the matrices: $A=\begin{bmatrix} 1 & 2 & -1 & 3\\ 5 & 1 & 1 & 3 \end{bmatrix}$, $B=\begin{bmatrix} 0 & 1 & 2 & -1\\ 1 & 1 & 0 & 0 \end{bmatrix}$ and $C=\begin{bmatrix} 2 & 1\\ 1 & -1 \end{bmatrix}$. The matrix $M=\dfrac{1}{3} \left( BA^T\right)^T - C$ is:

# Operation

Training
13

#### Inverse of a matrix

It is known that a sequence of elementary operations on rows transforms the matrix $[A|I_3]$ into $\begin{bmatrix}1&3&0&|&1&2&3\\0&1&2&|&1&0&2\\0&0&1&|&2&3&1\\\end{bmatrix}$. The $A^{-1}$ matrix is:

# Inverse

Learning
13

#### 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