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