## All Tasks

#### System of linear equations

Given the following matrices $A$ and $B$, find the matrix $X$ such that $AX=B$:
$A=\begin{bmatrix}
0 & 1 & 3 \\
-1 & 2 & 0 \\
1 & -1 & 0
\end{bmatrix}$
$B=\begin{bmatrix}
6 \\ 0 \\ 0
\end{bmatrix}$
(In the answer, fill the spaces that are not entries of the matrix X with a dot, ".".)

# System of linear equations

# Operation

Training

13

#### Chess board modulation

Consider the chess board represented in the figure. Modulating the situation by a numeric matrix, A, in which pawns are represented by the number -1, kings by the number 1, queens by the number 2, rooks by the number 3, and empty spaces with zeros, fill in the blanks of matrix A. What is the dimension of the matrix? If the black queen moves 5 spaces vertically and 3 spaces to the left, what is the position in the matrix of the number representing this piece after the moves?

# Matrices

Modeling

13

#### Matrix add 2

Compute the value of the constants, so that the equality is verified:
$\begin{bmatrix}
1& 0 &a\\ 0& b &-2
\end{bmatrix}-3
\begin{bmatrix}
2& c &-4\\ 0&\frac{1}{3} &d
\end{bmatrix}=
\begin{bmatrix}
e& 0 &8\\ 0& 2&7
\end{bmatrix}$

# Operation

Learning

13

#### 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