Question

Find the rank of the following matrices

`((-1, 2, -2),(4, -3, 4),(-2, 4, -4))`

Answer 1

Let A = `((-1, 2, -2),(4, -3, 4),(-2, 4, -4))`

Order of A is 3 × 3

∴ p(A) ≤ 3

Consider the third order minor = `|(-1, 2, -2),(4, -3, 4),(-2, 4, -4)|`

= – 1(12 – 16) – 2(– 16 + 8) – 2(16 – 6)

= – 1(– 4) – 2(– 8) – 2(10)

= 4 + 16 – 20

= 0

Since the third order minor vanishes, therefore

∴ p(A) ≠ 3

Consider a second order minor = `|(-1, 2),(4, -3)|`

= 3 – 8

= – 5 ≠ 0

There is a minor of order 2, which is not zero.
∴ p(A) = 2