Mathematics

Problem

3\times3

matrix

A=\begin{pmatrix}1 & 2 & 1\\ 0 & 1 & -1\\ 0 & 0 & 1\end{pmatrix}

is to be inverted by finding a matrix

B=\begin{pmatrix}a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33}\end{pmatrix}

that satisfies

AB=BA=I_3

. (i) Using

AB=I_3

, set up the nine linear equations in the nine unknown entries of

B

. (ii) Solve this system to obtain the explicit numerical value of each entry of

B

. (iii) Verify that the matrix found in part (ii) also satisfies

BA=I_3

. (iv) Comment on the uniqueness of the inverse of

A

in the light of your calculations.