Mathematics

Problem

Let

A=\begin{bmatrix}2 & 3\\ 4 & 6\end{bmatrix}

be a fixed

2\times 2

matrix. (a) Determine, in terms of variables

x, y, z, w

, all

2\times 2

matrices

B=\begin{bmatrix}x & y\\ z & w\end{bmatrix}

that satisfy

AB=O_{2\times 2}

but are themselves non-zero. (b) Using your result from part (a), construct one such matrix

B

whose entries are all integers. (c) Verify explicitly that the pair

(A,B)

you have chosen in part (b) indeed produces the zero matrix when multiplied in the order

AB