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=begin{bmatrix}2 & 0 &-11 & 4&35&4&2 end{bmatrix} text{ and } F=begin{bmatrix}2 & -1 &00 & 1&12&0&3 end{bmatrix} a) Show that A,B,C,D and F are invertible matrices. b) Solve the following equations for the unknown matrix X. (i) AX^T=BC^3 (ii) A^{-1}(X-T)^T=(B^{-1})^T (iii) XF=F^{-1}-D^T

Question
Matrices
asked 2020-12-16
Consider the matrices
\(A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix},B=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix},C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix},D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix} \text{ and } F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}\)
a) Show that A,B,C,D and F are invertible matrices.
b) Solve the following equations for the unknown matrix X.
(i) \(AX^T=BC^3\)
(ii) \(A^{-1}(X-T)^T=(B^{-1})^T\)
(iii) \(XF=F^{-1}-D^T\)

Answers (1)

2020-12-17
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
\(A=\begin{bmatrix}1 & -1 \\0 & 1 \end{bmatrix}\)
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
\(A=\begin{bmatrix}2 & 3 \\1 & 5 \end{bmatrix}\)
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
\(C=\begin{bmatrix}1 & 0 \\0 & 8 \end{bmatrix}\)
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
\(D=\begin{bmatrix}2 & 0 &-1\\1 & 4&3\\5&4&2 \end{bmatrix}\)
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
\(F=\begin{bmatrix}2 & -1 &0\\0 & 1&1\\2&0&3 \end{bmatrix}\)
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.
0

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