## All Tasks

#### Matrix operations 2

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

# Operation

Training

13

#### Matrix equation 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:
$A (I-X^T)^T = (X^{-1} B)^{-1}$

# Equations

Training

13

#### Matrix equation 2

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:
$(A^{-1}X^{-1}B)^{-1}=(A^T+B)^T$

# Equations

Learning

13

#### 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