Question

IF A=(ab b2)
(-a2 -ab)
show that a2=0 matrices

Answer 1

By computing the matrix product, we have

`A^2 = \left[\begin{array}{cc}ab&b^2\\-a^2&-ab\end{array}\right] \cdot\left[\begin{array}{cc}ab&b^2\\-a^2&-ab\end{array}\right]=\left[\begin{array}{cc}a^2b^2-a^2b^2&ab^3-ab^3\\-a^3b+a^3b&-a^2b^2+a^2b^2\end{array}\right]`

and as you can see, all the entries of this matrix are terms of the form
`x-x`, so the matrix is composed by nothing but zeroes.