Question

In matrix theory, many of the familiar properties of the real number system are not valid. If a and b are real numbers, then ab = 0 implies that a = 0 or b = 0. Find a matrix B such that AB = 0 where A = 1 0 0 0 ≠ 0 and B ≠ 0.

Answer 1

B=
`\left[\begin{array}{ccc}0&0\\0&1\\\end{array}\right]`

Explanation:

Let's do the multiplication AB.

If A=
`\left[\begin{array}{ccc}1&0\\0&0\\\end{array}\right]`

then the first row of A is= (1 0) by the first column of B= (0 0) is equal to zero.

the first row of A is= (1 0) by the second column of B= (0 1) is equal to zero too because 1.0+0.1=0.

the second row of A is= (0 0) by any colum of B is equal to zero too.

So we have found an example that works!