Question

Find the rank of the following matrices

`((1, -2, 3, 4),(-2, 4, -1, -3),(-1, 2, 7, 6))`

Answer 1

Let A = `((1, -2, 3, 4),(-2, 4, -1, -3),(-1, 2, 7, 6))`

Order of A is 3 × 4

∴ p(A) ≤ 3

Consider the third order minor

`|(1, -2, 3),(-2, 4, -1),(-1, 2, 7)|` = 1(28 + 2) + 2(−14 – 1) + 3(−4 + 4)

= 1(30) + 2(−15) + 3(0)

= 0

`|(-2, 3, 4),(4, -1, -3),(2, 7, 6)|` = – 2(– 6 + 21) – 3(24 + 6) + 4(28 + 2)

= – 2(15) – 3(30) + 4(30)

= 0

`|(1, 3, 4),(-2, -1, -3),(-1, 7, 6)|` = 1(– 6 + 21) – 3(– 12 – 3) + 4(– 4 – 1)

= 1(15) – 3(–15) + 4(–15)

= 15 + 45 – 60

= 0

`|(1, -2, 4),(-2, 4, -3),(-1, 2, 6)|` = 1(24 + 6) + 2(–12 – 3) + 4(– 4 + 4)

= 1(30) + 2(–15) + 4(0)

= 30 – 30

= 0

Since all third order minors vanishes, ρ(A) ≠ 3

Now, let us consider the second order minors.

Consider one of the second order minors

`|(-2, 3),(4, -1)|` = 2 – 12

= –10 ≠ 0

There is a minor of order 2 which is not zero

∴ p(A) = 2