# Consider the matrices A=begin{bmatrix}1 & -1 0 & 1 end{bmatrix},B=begin{bmatrix}2 & 3 1 & 5 end{bmatrix},C=begin{bmatrix}1 & 0 0 & 8 end{bmatrix},D=be

Consider the matrices

a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) $A{X}^{T}=B{C}^{3}$
(ii) ${A}^{-1}\left(X-T{\right)}^{T}=\left({B}^{-1}{\right)}^{T}$
(iii) $XF={F}^{-1}-{D}^{T}$
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estenutC

Step 1
(a) A matrix S is invertible if the determinant of the matrix is not 0
That is , $det\left(S\right)\ne 0$
Step 2
Consider the matrix A
$A=\left[\begin{array}{cc}1& -1\\ 0& 1\end{array}\right]$
Obtain the detrminant of A
$det\left(A\right)=1\cdot \left(1\right)-\left(-1\right)\cdot 0$
$=1+0$
$=1$
$\ne 0$
Hence , A is invertible
Step 3
Consider the matrix B
$A=\left[\begin{array}{cc}2& 3\\ 1& 5\end{array}\right]$
Obtain the detrminant of B
$det\left(B\right)=2\cdot \left(5\right)-\left(3\right)\cdot 1$
$=10-3$
$=7$
$\ne 0$
Hence , B is invertible
Step 4
Consider the matrix C
$C=\left[\begin{array}{cc}1& 0\\ 0& 8\end{array}\right]$
Obtain the detrminant of A
$det\left(C\right)=1\cdot \left(8\right)-\left(0\right)\cdot 0$
$=8$
$\ne 0$
Hence , C is invertible
Step 5
Consider the matrix D
$D=\left[\begin{array}{ccc}2& 0& -1\\ 1& 4& 3\\ 5& 4& 2\end{array}\right]$
Obtain the detrminant of D
$det\left(D\right)=2\left[8-12\right]-0\left[2-15\right]-1\left[4-20\right]$
$=2\cdot \left(-4\right)-\left(-16\right)$
$=8$
$\ne 0$
Hence , D is invertible
Step 6
Consider the matrix F
$F=\left[\begin{array}{ccc}2& -1& 0\\ 0& 1& 1\\ 2& 0& 3\end{array}\right]$
Obtain the detrminant of F
$det\left(F\right)=2\left[3-0\right]+1\left[0-2\right]+0\left[0-2\right]$
$=2\cdot \left(3\right)+\left(-2\right)$
$=6-2=4$
$\ne 0$
Hence , F is invertible
Thus, A, B, C, D and F are invertible matrices.

Jeffrey Jordon