Advertisements
Advertisements
प्रश्न
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)]`
उत्तर
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)
APPEARS IN
संबंधित प्रश्न
Without expanding the determinant, prove that `|("s", "a"^2, "b"^2 + "c"^2),("s", "b"^2, "c"^2 + "a"^2),("s", "c"^2, "a"^2 + "b"^2)|` = 0
Prove that `|("a"^2, "bc", "ac" + "c"^2),("a"^2 + "ab", "b"^2, "ac"),("ab", "b"^2 + "bc", "c"^2)| = 4"a"^2"b"^2"c"^2`
Prove that `|(sec^2theta, tan^2theta, 1),(tan^2theta, sec^2theta, -1),(38, 36, 2)|` = 0
If `|("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)|` = 0, prove that a, b, c are in G. P or α is a root of ax2 + 2bx + c = 0
If A is a Square, matrix, and |A| = 2, find the value of |A AT|
Determine the roots of the equation `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)|` = 0
Show that `|("b" + "c", "a" - "c", "a" - "b"),("b" - "c", "c" + "a", "b" - "a"),("c" - "b", "c" - "a", "a" + b")|` = 8abc
Identify the singular and non-singular matrices:
`[(1, 2, 3),(4, 5, 6),(7, 8, 9)]`
Identify the singular and non-singular matrices:
`[(2, -3, 5),(6, 0, 4),(1, 5, -7)]`
Identify the singular and non-singular matrices:
`[(0, "a" - "b", "k"),("b" - "a", 0, 5),(-"k", -5, 0)]`
Determine the values of a and b so that the following matrices are singular:
A = `[(7, 3),(-2, "a")]`
Determine the values of a and b so that the following matrices are singular:
B = `[("b" - 1, 2, 3),(3, 1, 2),(1, -2, 4)]`
Choose the correct alternative:
The value of x, for which the matrix A = `[("e"^(x - 2), "e"^(7 + x)),("e"^(2 + x), "e"^(2x + 3))]` is singular
Choose the correct alternative:
If x1, x2, x3 as well as y1, y2, y3 are in geometric progression with the same common ratio, then the points (x1, y1), (x2, y2), (x3, y3) are
Find the area of the triangle with vertices at the point given is (1, 0), (6, 0), (4, 3).
Let a, b, c, d be in arithmetic progression with common difference λ. If `|(x + a - c, x + b, x + a),(x - 1, x + c, x + b),(x - b + d, x + d, x + c)|` = 2, then value of λ2 is equal to ______.
For f(x)= `ℓn|x + sqrt(x^2 + 1)|`, then the value of`g(x) = (cosx)^((cosecx - 1))` and `h(x) = (e^x - e^-x)/(e^x + e^-x)`, then the value of `|(f(0), f(e), g(π/6)),(f(-e), h(0), h(π)),(g((5π)/6), h(-π), f(f(f(0))))|` is ______.
Let S = `{((a_11, a_12),(a_21, a_22)): a_(ij) ∈ {0, 1, 2}, a_11 = a_22}`
Then the number of non-singular matrices in the set S is ______.
