Advertisements
Advertisements
प्रश्न
If A = `[(1, 2),(3, 4)]` verify that A (adj A) = (adj A) A = |A| I
उत्तर
For A = `[(1, 2),(3, 4)]`
A11 = (–1)1 + 1 (4) = 4
A12 = (–1)1 + 2 (3) = –3
A21 = (–1)2 + 1 (2) = –2
A22 = (–1)2 + 2 (1) = 1
adj A = `[(A_11, A_21),(A_12, A_22)]`
= `[(4, -2),(-3, 1)]`
∴ A(adj A) = `[(1, 2),(3, 4)][(4, -2),(-3, 1)]`
= `[(4 - 6, -2 +2),(12 -12, -6 + 4)]`
= `[(-2, 0),(0, -2)]` ...(i)
(adj A) . A = `[(4, -2),(-3, 1)][(1, 2),(3, 4)]`
= `[(4 - 6, 8 - 8),(-3 + 3, -6 + 4)]`
= `[(-2, 0),(0, -2)]` ...(ii)
and |A| I = `[(1, 2),(3, 4)][(1, 0),(0, 1)]` = 4 -6 = -2
= `(-2) [(1, 0),(0, 1)]`
= `[(-2, 0),(0, -2)]` ...(iii)
From (i), (ii) and (iii) we get,
A(adj A) = (adj A) A = |A| I
APPEARS IN
संबंधित प्रश्न
Find the co-factor of the element of the following matrix.
`[(1,-1,2),(-2,3,5),(-2,0,-1)]`
Find the inverse of the following matrix.
`[(1,2),(2,-1)]`
Find the inverses of the following matrices by the adjoint method:
`[(1,2,3),(0,2,4),(0,0,5)]`
Find AB, if A = `((1,2,3),(1,-2,-3))` and B = `((1,-1),(1,2),(1,-2))`. Examine whether AB has inverse or not.
Choose the correct answer from the given alternatives in the following question:
If A = `[("cos"alpha, - "sin"alpha,0),("sin"alpha,"cos"alpha,0),(0,0,1)]` where α ∈ R, then [F(α)]-1 is
Choose the correct answer from the given alternatives in the following question:
The inverse of A = `[(0,1,0),(1,0,0),(0,0,1)]` is
Find the inverse of the following matrices by the adjoint method `[(3, -1),(2, -1)]`.
Adjoint of `[(2, -3),(4, -6)]` is _______
Choose the correct alternative.
If a 3 x 3 matrix B has it inverse equal to B, thenB2 = _______
If A = `[(1, 2),(-3, -1)], "B" = [(-1, 0),(1, 5)]`, then AB =
Fill in the blank :
If A = `[(2, 1),(1, 1)] "and" "A"^-1 = [(1, 1),(x, 2)]`, then x = _______
State whether the following is True or False :
If A and B are conformable for the product AB, then (AB)T = ATBT.
Find inverse of the following matrices (if they exist) by elementary transformations :
`[(1, -1),(2, 3)]`
`cos theta [(cos theta, sin theta),(-sin theta, cos theta)] + sin theta [(sin theta, - cos theta),(cos theta, sin theta)]` = ______
If A = `[("a", "b"),("c", "d")]` then find the value of |A|−1
If A(α) = `[(cos alpha, sin alpha),(-sin alpha, cos alpha)]` then prove that A2(α) = A(2α)
If A = `[(0, 3, 3),(-3, 0, -4),(-3, 4, 0)]` and B = `[(x),(y),(z)]`, find the matrix B'(AB)
If A = `[(2, 4),(1, 3)]` and B = `[(1, 1),(0, 1)]` then find (A−1 B−1)
If A = `[(-4, -3, -3),(1, 0, 1),(4, 4, 3)]`, find adj (A).
State whether the following statement is True or False:
Inverse of `[(2, 0),(0, 3)]` is `[(1/2, 0),(0, 1/3)]`
The value of Minor of element b22 in matrix B = `[(2, -2),(4, 5)]` is ______
Find the inverse of the following matrix:
`[(1,-1),(2,3)]`
Find the inverse of the following matrix:
`[(-3,-5,4),(-2,3,-1),(1,-4,-6)]`
If A = `[(2,3),(1,-6)]` and B = `[(-1,4),(1,-2)]`, then verify adj (AB) = (adj B)(adj A)
Show that the matrices A = `[(2,2,1),(1,3,1),(1,2,2)]` and B = `[(4/5,(-2)/5,(-1)/5),((-1)/5,3/5,(-1)/5),((-1)/5,(-2)/5,4/5)]` are inverses of each other.
If A = `[(3,7),(2,5)]` and B = `[(6,8),(7,9)]`, then verify that (AB)-1 = B-1A-1
Solve by matrix inversion method:
3x – y + 2z = 13; 2x + y – z = 3; x + 3y – 5z = - 8
Solve by matrix inversion method:
2x – z = 0; 5x + y = 4; y + 3z = 5
A sales person Ravi has the following record of sales for the month of January, February and March 2009 for three products A, B and C. He has been paid a commission at fixed rate per unit but at varying rates for products A, B and C.
| Months | Sales in units | Commission | ||
| A | B | C | ||
| January | 9 | 10 | 2 | 800 |
| February | 15 | 5 | 4 | 900 |
| March | 6 | 10 | 3 | 850 |
Find the rate of commission payable on A, B and C per unit sold using matrix inversion method.
adj (AB) is equal to:
The matrix M = `[(0,1,2),(1,2,3),(3,1,1)]` and its inverse is N = [nij]. What is the element n23 of matrix N?
If A = `[(p/4, 0, 0), (0, q/5, 0), (0, 0, r/6)]` and `"A"^-1 = [(1/4, 0, 0), (0, 1/5, 0), (0, 0, 1/6)]`, then p + q + r = ______
If A = `[(0, -1, 0), (1, 0, 0), (0, 0, -1)]`, then A-1 is ______
If A and Bare square matrices of order 3 such that |A| = 2, |B| = 4, then |A(adj B)| = ______.
If A = `[(2, -3), (3, 5)]`, then |Adj A| is equal to ______
If A = `[(5, -4), (7, -5)]`, then 3A-1 = ______
If A is a solution of x2 - 4x + 3 = 0 and `A=[[2,-1],[-1,2]],` then A-1 equals ______.
The matrix `[(lambda, 1, 0),(0, 3, 5),(0, -3, lambda)]` is invertible ______.
The inverse of the matrix A = `[(3, 0, 0),(0, 4, 0),(0, 0, 5)]` is ______.
Find the inverse of the matrix A by using adjoint method.
where A = `[(-3, -1, 1),(0, 0, 1),(-15, 6, -6)]`
If matrix A = `[(3, -2, 4),(1, 2, -1),(0, 1, 1)]` and A–1 = `1/k` (adj A), then k is ______.
The number of solutions of equation x2 – x3 = 1, – x1 + 2x3 = 2, x1 – 2x2 = 3 is ______.
If A = `[(1, 2, 3),(-1, 1, 2),(1, 2, 4)]` then (A2 – 5A)A–1 = ______.
Matrix A = `[(1, 2, 3),(1, 1, 5),(2, 4, 7)]` then the value of a31 A31 + a32 A32 + a33 A33 is ______.
If A = `[(cos α, sin α),(-sin α, cos α)]`, then find α satisfying `0 < α < π/2`, when A + AT = `sqrt(2) l_2` where AT is transpose of A.
if `A = [(2,-1,1),(-1,2,-1),(1,-1,2)]` then find A−1 by the adjoint method.
If A = `[(1, 2, 4),(4, 3, -2),(1, 0, -3)]`. Show that A–1 exists and find A–1 using column transformation.
