Solve the system of equations using matrices. Use the Gaussian elimination method with​ back-substitution. x+4y=0 x+5y+z=1 5x-y-z=79

Find a basis for the space of $2\times 2$ diagonal matrices.
$\text{Basis }=\{\left[\begin{array}{cc}& \\ & \end{array}\right],\left[\begin{array}{cc}& \\ & \end{array}\right]\}$

Let B be a 4x4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product of ABC (same B) of three matrices.

Perform the indicated matrix operations B - A given that A, B, and C are defined as follows. If an operation is not defined, state the reason.
$A=\left[\begin{array}{cc}4& 0\\ -3& 5\\ 0& 1\end{array}\right]B=\left[\begin{array}{cc}5& 1\\ -2& -2\end{array}\right]C=\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$

Determine whether the statement, $(A+B{)}^{-1}=A^{-1}+B^{-1}$, assuming A, B, and A + B are invertible, is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

Given the matrices
$A=\left[\begin{array}{cc}-1& 3\\ 2& -1\\ 3& 1\end{array}\right]\text{ and }B=\left[\begin{array}{cc}0& -2\\ 1& 3\\ 4& -3\end{array}\right]$ find the $3\times 2$ matrix X that is a solution of the equation. 8X+A=B

Step 1
Solve absolute value inequality.
${\displaystyle -4\left|1-x\right|<-16}$

Given n>1 , show that the set of all $1\times n$ matrices of positive integers in countable.