Question

Given, matrix `A = [(x, 1),(y, 2)]` and `B = [(x),(x - 2)]` such that AB is a null matrix. Find:

1. order of the null matrix. 
2. possible values of x and y.

Answer 1

Given: `A = [(x, 1),(y, 2)]` and `B = [(x),(x - 2)]` such that AB is a null zero matrix. Null matrix = all entries 0.

Step-wise calculation:

1. Determine order of AB:

A is 2 × 2 and B is 2 × 1

So, AB is of order 2 × 1 (two rows, one column).

2. Compute AB and set it equal to the zero column:

AB = `[(x, 1), (y, 2)] [(x), (x - 2)]` 

= `[((x xx x) + (1 xx (x - 2))), ((y xx x) + (2 xx (x - 2)))]`

= `[(x^2 + x - 2), (xy + 2x - 4)]` 

= `[(0), (0)]`

3. Solve the first equation:

x² + x – 2 = 0

Factor: (x + 2)(x – 1) = 0

So x = 1 or x = –2.

4. For each x, solve the second equation xy + 2x – 4 = 0 for y:

If x = 1: y(1) + 2(1) – 4 = 0

⇒ y + 2 – 4 = 0

⇒ y = 2

If x = –2: y(–2) + 2(–2) – 4 = 0 

⇒ –2y – 4 – 4 = 0 

⇒ –2y – 8 = 0 

⇒ y = –4