Advertisements
Advertisements
प्रश्न
If A = `[(2, 3),(1, 2)]`, B = `[(1, 0),(3, 1)]`, find AB and (AB)−1
Advertisements
उत्तर
AB = `[(2, 3),(1, 2)][(1, 0),(3, 1)]``
= `[(2 + 9, 0 + 3),(1 + 6, 0 + 2)]`
= `[(11, 3),(7, 2)]`
Now, |AB| = `|(11, 3),(7, 2)|`
= 22 − 21
= 1 ≠ 0
∴ (AB)−1 exists.
Consider, (AB)(AB)−1 = I
∴ `[(11, 3),(7, 2)]` (AB)−1 = `[(1, 0),(0, 1)]`
Applying R1 → 2R1,
`[(22, 6),(7, 2)]` (AB)−1 = `[(2, 0),(0, 1)]`
Applying R1 → R1 − 3R2,
`[(1, 0),(7, 2)]` (AB)−1 = `[(2, -3),(0, 1)]`
Applying R2 → R2 − 7R1,
`[(1, 0),(0, 2)]` (AB)−1 = `[(2, -3),(-14, 22)]`
Applying R2 → `(1/2)` R2
`[(1, 0),(0, 1)]` (AB)−1 = `[(2, -3),(-7, 11)]`
∴ (AB)−1 = `[(2, -3),(-7, 11)]` .......(1)
|A| = `|(2,3),(1,2)| = 4 - 3 = 1 ne 0`
∴ A−1 exists.
Consider, AA−1 = I
∴ `[(2, 3),(1, 2)]` A−1 = `[(1, 0),(0, 1)]`
Applying R1 ↔ R2,
`[(1, 2),(2, 3)]` A−1 = `[(0, 1),(1, 0)]`
Aplying 2R2 → R1,
`[(1, 2),(0, -1)]` A−1 = `[(0, 1),(1, -2)]`
Applying (−1)R2 ↔ R1,
`[(1, 2),(0, 1)]` A−1 = `[(0, 1),(-1, 2)]`
Applying R1 ↔ R2,
`[(1, 0),(0, 1)]` A−1 = `[(2, -3),(-1, 2)]`
∴ A−1 = `[(2, -3),(-1, 2)]`
|B| = `|(1, 0),(3, 1)|`
= 1 − 0
= 1 ≠ 0
∴ B−1 exists.
Consider, BB−1 = I
∴ `[(1, 0),(3, 1)]` B−1 = `[(1, 0),(0, 1)]`
Applying R2 ↔ 3R1,
`[(1, 0),(0, 1)]` A−1 = `[(1, 0),(-3, 1)]`
∴ B−1 = `[(1, 0),(-3, 1)]`
∴ B−1. A−1 = `[(1, 0),(-3, 1)],[(2, -3),(-1, 2)]`
= `[(2 - 0, -3+0),(-6-1, 9+2)]`
= `[(2, -3),(-7,11)]` .......(2)
From (1) and (2), (AB)−1 = B−1. A−1
APPEARS IN
संबंधित प्रश्न
Apply the given elementary transformation of the following matrix.
B = `[(1, -1, 3),(2, 5, 4)]`, R1→ R1 – R2
Apply the given elementary transformation of the following matrix.
A = `[(5,4),(1,3)]`, C1↔ C2; B = `[(3,1),(4,5)]` R1↔ R2.
What do you observe?
Apply the given elementary transformation of the following matrix.
A = `[(1,2,-1),(0,1,3)]`, 2C2
B = `[(1,0,2),(2,4,5)]`, −3R1
Find the addition of the two new matrices.
Apply the given elementary transformation of the following matrix.
A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R3 and then C3 + 2C2
Apply the given elementary transformation of the following matrix.
A = `[(1,-1,3),(2,1,0),(3,3,1)]`, 3R3 and then C3 + 2C2
and A = `[(1,-1,3),(2,1,0),(3,3,1)]`, C3 + 2C2 and then 3R3
What do you conclude?
Apply the given elementary transformation of the following matrix.
Convert `[(1,-1),(2,3)]` into an identity matrix by suitable row transformations.
If A = `((1,0,0),(2,1,0),(3,3,1))`, then reduce it to I3 by using column transformations.
If A = `[(2,1,3),(1,0,1),(1,1,1)]`, then reduce it to I3 by using row transformations.
Check whether the following matrix is invertible or not:
`[(1,0),(0,1)]`
Check whether the following matrix is invertible or not:
`((1,1),(1,1))`
Check whether the following matrix is invertible or not:
`((1,2),(3,3))`
Check whether the following matrix is invertible or not:
`((2,3),(10,15))`
Check whether the following matrix is invertible or not:
`[(cos theta, sin theta),(-sin theta, cos theta)]`
Check whether the following matrix is invertible or not:
`(("sec" theta , "tan" theta),("tan" theta,"sec" theta))`
Check whether the following matrix is invertible or not:
`[(3,4,3),(1,1,0),(1,4,5)]`
Check whether the following matrix is invertible or not:
`((1,2,3),(2,-1,3),(1,2,3))`
If A = `[("x",0,0),(0,"y",0),(0,0,"z")]` is a non-singular matrix, then find A−1 by using elementary row transformations. Hence, find the inverse of `[(2,0,0),(0,1,0),(0,0,-1)]`
If A = `[(1,2),(3,4)]` and X is a 2 × 2 matrix such that AX = I, find X.
Find the inverse of A = `[("cos" theta, -"sin" theta, 0),("sin" theta, "cos" theta, 0),(0,0,1)]` by elementary column transformations.
If A = `[(2,3),(1,2)]`, B = `[(1,0),(3,1)]`, find AB and (AB)-1 . Verify that (AB)-1 = B-1.A-1.
If A = `[(4,5),(2,1)]`, show that `"A"^-1 = 1/6("A" - 5"I")`.
Find the matrix X such that AX = B, where A = `[(1,2),(-1,3)]` and B = `[(0,1),(2,4)]`
If A = `[(1,1),(1,2)], "B" = [(4,1),(3,1)]` and C = `[(24,7),(31,9)]`, then find the matrix X such that AXB = C
Find the inverse of A = `[(1,0,1),(0,2,3),(1,2,1)]` by using elementary column transformations.
Show with the usual notation that for any matrix A = `["a"_"ij"]_(3xx3) "is" "a"_11"A"_11 + "a"_12"A"_12 + "a"_13"A"_13 = |"A"|`
If A = `[(1,0,1),(0,2,3),(1,2,1)]` and B = `[(1,2,3),(1,1,5),(2,4,7)]`, then find a matrix X such that XA = B.
Find the inverse of the following matrix (if they exist).
`[(1,3,-2),(-3,0,-5),(2,5,0)]`
Choose the correct answer from the given alternatives in the following question:
If A = `[(1,2),(3,4)]` , adj A = `[(4,"a"),(-3,"b")]`, then the values of a and b are
Choose the correct answer from the given alternatives in the following question:
The inverse of `[(0,1),(1,0)]` is
Choose the correct answer from the given alternatives in the following question:
If A = `[(1,2),(2,1)]` and A(adj A) = k I, then the value of k is
The element of second row and third column in the inverse of `[(1, 2, 1),(2, 1, 0),(-1, 0, 1)]` is ______.
Find A−1 using column transformations:
A = `[(5, 3),(3, -2)]`
Find A−1 using column transformations:
A = `[(2, -3),(-1, 2)]`
Find the matrix X such that `[(1, 2, 3),(2, 3, 2),(1, 2, 2)]` X = `[(2, 2, -5),(-2, -1, 4),(1, 0, -1)]`
Find the inverse of A = `[(2, -3, 3),(2, 2, 3),(3, -2, 2)]` by using elementary row transformations.
If A = `[(3, -1),(4, 2)]`, B = `[(2),(-1)]`, find X such that AX = B.
