## All Tasks

#### Electric circuits and system of linear equations

Compute the currents in the electrical circuit shown in the figure.

# System of linear equations

Modeling

13

#### SLE as a matrix equation

A system of linear equations can be represented as the matrix equation $A X = B$. Consider the system of linear equations
$$
\left\{
\begin{array}{l}
2x+3y+2z=1\\
x+4y-z=0\\
5x+z=9
\end{array}
\right.
$$
The matrices $A$, $X$ and $B$ of the system are:

# Equations

# System of linear equations

Training

13

#### Matrix equation 0.2

Let $A$, $B$, $C$ and $X$ be matrices with real values, and suppose that all the matrices involved have an inverse and the operations involved are all possible. Solve in order of $X$ the following matrix equation:
$( X A +B )^T = C$

# Equations

Learning

13

#### Matrix equation 0.1

Let $A$, $B$, and $X$ be matrices with real values, and suppose that all the matrices involved have an inverse and the operations involved are all possible. Solve in order of $X$ the following matrix equation:
$A X = B$

# Equations

Learning

13

#### Power of a matrix 1

Consider the identity matrix $I_4$ and the matrix
$$A=\begin{bmatrix}
0&1&1&1\\1& 0&1&1\\1&1&0&1\\1&1&1&0
\end{bmatrix}$$
Matrix $A^2$ is a linear combination of matrices A and $I_4$,
ie,
$\exists x,y\in R: A^2 = x A + y I_4$
Which of the following statements is true?

# Operation

Reasoning

13

#### Matrices multiplication 0

Let $A=\begin{bmatrix} 4&-2&5\\2&6&0\\3&3&3\end{bmatrix}$ and $B=\begin{bmatrix} 1&0&0\\0&2&-3\\6&-6&2\end{bmatrix}$
be real matrices $M_{3\times 3}$.
The matrix $AB$ is:

# Operation

Training

13