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प्रश्न
Find the rank of the following matrices
`((1, -2, 3, 4),(-2, 4, -1, -3),(-1, 2, 7, 6))`
उत्तर
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
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