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

Question

Consider the matrices

A = [\begin{matrix} 1 & - 1 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 2 & 3 \\ 1 & 5 \end{matrix}], C = [\begin{matrix} 1 & 0 \\ 0 & 8 \end{matrix}], D = [\begin{matrix} 2 & 0 & - 1 \\ 1 & 4 & 3 \\ 5 & 4 & 2 \end{matrix}] and F = [\begin{matrix} 2 & - 1 & 0 \\ 0 & 1 & 1 \\ 2 & 0 & 3 \end{matrix}]

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} (X - T)^{T} = (B^{- 1})^{T}

(iii)

X F = F^{- 1} - D^{T}

Answer 1

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