Matrices and matrix operations

Given the matrices

$$A=\left(\begin{array}{ccc}1& 0& -1\\ 2& 3& 1\end{array}\right),B=\left(\begin{array}{ccc}0& 1& 1\\ 1& -1& 2\end{array}\right),$$

compute $2A-3B$.

Solution available to those who have unlocked the subject.

Given the matrices

$$A=\left(\begin{array}{ccc}1& -1& 2\\ 0& 3& 1\end{array}\right),B=\left(\begin{array}{cc}5& 3\\ 1& 6\\ 7& 5\end{array}\right),$$

find the matrix $X$ such that $3X+2A^T=B$.

Solution available to those who have unlocked the subject.

Given the matrices

$$A=\left(\begin{array}{cc}1& 2\\ 0& -1\\ 3& 1\end{array}\right),B=\left(\begin{array}{ccc}2& 0& 1\\ 1& -1& 0\end{array}\right),$$

compute, if defined, the products $AB$ and $BA$.

Solution available to those who have unlocked the subject.

Determine whether the matrices

$$A=\left(\begin{array}{cc}1& 2\\ 0& 3\end{array}\right),B=\left(\begin{array}{cc}3& -2\\ 0& 1\end{array}\right)$$

commute.

Solution available to those who have unlocked the subject.

Reduce the matrix

$$A=\left(\begin{array}{ccc}2& 4& -2\\ 1& 3& 0\\ 3& 5& -4\end{array}\right)$$

to row echelon form and indicate how many pivots are obtained and in which columns.

Solution available to those who have unlocked the subject.

Find the reduced row echelon form of the matrix

$$A=\left(\begin{array}{cccc}1& 2& 0& 1\\ 2& 4& 1& 4\\ 1& 2& 1& 3\end{array}\right).$$

Solution available to those who have unlocked the subject.

Compute with the Gauss-Jordan method the inverse of the matrix

$$A=\left(\begin{array}{cc}2& 5\\ 1& 3\end{array}\right).$$

Solution available to those who have unlocked the subject.

Compute with the Gauss-Jordan method the inverse of the matrix

$$A=\left(\begin{array}{ccc}1& 0& 2\\ 2& 1& 3\\ 1& 1& 2\end{array}\right).$$

Solution available to those who have unlocked the subject.

Find the values of the real parameter $k$ for which the matrix

$$A_k=\left(\begin{array}{ccc}1& 1& k\\ 1& k& 1\\ k& 1& 1\end{array}\right)$$

is invertible.

Solution available to those who have unlocked the subject.

Compute $A^n$ for every integer $n\ge 1$, where

$$A=\left(\begin{array}{cc}1& 1\\ 0& -1\end{array}\right).$$

Solution available to those who have unlocked the subject.

Solve the matrix equation $AX=B$, where

$$A=\left(\begin{array}{cc}2& 1\\ 5& 3\end{array}\right),B=\left(\begin{array}{cc}1& 0\\ 2& 1\end{array}\right).$$

Solution available to those who have unlocked the subject.

Find all matrices $X\in M_2(\mathbb{R})$ that commute with

$$A=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).$$

Solution available to those who have unlocked the subject.

Given the matrix

$$A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right),$$

verify that $A^2-4A+3I_2=O$ and use it to find $A^{-1}$ without row reduction.

Solution available to those who have unlocked the subject.