All Tasks
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