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Question
Verify that det(AB) = (det A)(det B) for A = `[(4, 3, -2),(1, 0, 7),(2, 3, -5)]` and B = `[(1, 3, 3),(-2, 4, 0),(9, 7, 5)]`
Solution
Given A = `[(4, 3, -2),(1, 0, 7),(2, 3, -5)]`
B = `[(1, 3, 3),(-2, 4, 0),(9, 7, 5)]`
AB = `[(4, 3, -2),(1, 0, 7),(2, 3, -5)] [(1, 3, 3),(-2, 4, 0),(9, 7, 5)]`
= `[(4 - 6 - 18, 12 + 12 - 14, 12 + 0 - 10),(1 - 0 + 63,3 +0 + 49, 3 + 0 + 35),(2 - 6 - 45, 6 + 12 - 35, 6 + 0 - 25)]`
= `[(-20, 10, 2),(64, 52, 38),(-49, -1, -19)]`
det(AB) = |AB|
= `|(-20, 10, 2),(64, 52, 38),(-49, - 17, -19)|`
= `2 xx 2 |(-10, 5, 1),(32, 26, 19),(-49, -17, -19)|`
= `4|(-10, 5, 1),(32, 26, 19),(-17, -9, 0)| "R"_3 -> "R"_3 + "R"_2`
= 4 [– 10 (0 – 9 × 19) – 5(0 + 17 × 19) + 1(32 × 9 + 17 × 26)]
= 4[1710 – 5 × 323 + 288 + 442]
= 4[1710 – 1615 + 730]
= 4[2440 – 1615]
= 4 × 825
det (AB) = 3300 .......(1)
A = `[(4, 3, -2),(1, 0, 7),(2, 3, -5)]`
|A| = `[(4, 3, -2),(1, 0, 7),(2, 3, -5)]`
= 4(0 – 21) – 3(– 5 – 14) – 2(3 – 0)
= – 84 – 3 × – 19 – 6
= – 84 + 57 – 6
= – 90 + 57
det A = – 33 .......(2)
B = `[(1, 3, 3),(-2, 4, 0),(9, 7, 5)]`
|B| = `|(1, 3, 3),(-2, 4, 0),(9, 7, 5)|`
= 1(20 – 0) – 3(– 10 – 0) + 3(– 14 – 36)
= 20 + 30 + 3 × – 50
= 50 – 150
det A = – 100 .......(3)
From equations (2) and (3)
(det A)(det B) = – 33 × – 100
(detA)(det B) = 3300 ........(4)
From equations (1) and (4), we have
det(AB) = (det A)(det B)
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