Advertisements
Advertisements
Question
Find the adjoint of the following matrix.
`[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`
Solution
Let A = `[(1, -1, 2),(-2, 3, 5),(-2, 0, -1)]`
Now, M11 = `|(3, 5),(0, -1)|` = −3 − 0 = −3
∴ A11 = (−1)1+1(− 3) = −3
M12 = `|(-2, 5),(-2, -1)|` = 2 + 10 = 12
∴ A12 = (−1)1+2(12) = −12
M13 = `|(-2, 3),(-2, 0)|` = 0 + 6 = 6
∴ A13 = (− 1)1+3(6) = 6
M21 = `|(-1,2),(0,-1)|` = 1 − 0 = 1
∴ A21 = (−1)2+1(1) = −1
M22 = `|(1, 2),(-2, -1)|` = −1 + 4 = 3
∴ A22 = (−1)2+2(3) = 3
M23 = `|(1, -1),(-2, 0)|` = 0 − 2 = −2
∴ A23 = (−1)2+3(−2) = 2
M31 = `|(-1, 2),(3, 5)|` = −5 − 6 = −11
∴ A31 = (−1)3+1(−11) = −11
M32 = `|(1, 2),(-2, 5)|` = 5 + 4 = 9
∴ A32 = (−1)3+2(9) = −9
M33 = `|(1, -1),(-2, 3)|` = 3 − 2 = 1
∴ A33 = (−1)3+3(1) = 1
The co-factor matrix = `[("A"_11, "A"_12, "A"_13),("A"_21, "A"_22, "A"_23),("A"_31, "A"_32, "A"_33)]`
∴ adj A = `[(-3, -12, 6),(-1, 3, 2),(-11, -9, 1)]^"T"`
= `[(-3, -1, -11),(-12, 3, -9),(6, 2, 1)]`
APPEARS IN
RELATED QUESTIONS
The sum of three numbers is 6. If we multiply the third number by 3 and add it to the second number we get 11. By adding first and third numbers we get a number, which is double than the second number. Use this information and find a system of linear equations. Find these three numbers using matrices.
Find the inverse of the matrix `[(1 2 3),(1 1 5),(2 4 7)]` by adjoint method
Solve the following equations by the inversion method :
2x + 3y = - 5 and 3x + y = 3.
Find the adjoint of the following matrix.
`[(2,-3),(3,5)]`
Find the inverse of the following matrix (if they exist):
`((1,-1),(2,3))`
Choose the correct answer from the given alternatives in the following question:
For a 2 × 2 matrix A, if A(adj A) = `[(10,0),(0,10)]`, then determinant A equals
Choose the correct alternative.
If a 3 x 3 matrix B has it inverse equal to B, thenB2 = _______
A = `[(cos alpha, - sin alpha, 0),(sin alpha, cos alpha, 0),(0, 0, 1)]`, then A−1 is
A + I = `[(3, -2),(4, 1)]` then find the value of (A + I)(A − I)
If A = `[(-1),(2),(3)]`, B = `[(3, 1, -2)]`, find B'A'
If A = `[(6, 5),(5, 6)]` and B = `[(11, 0),(0, 11)]` then find A'B'
If A = `[(0, 1),(2, 3),(1, -1)]` and B = `[(1, 2, 1),(2, 1, 0)]`, then find (AB)−1
Complete the following activity to verify A. adj (A) = det (A) I.
Given A = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)]` then
|A| = 2(____) – 0(____) + ( ) (____)
= 6 – 0 – 5
= ______ ≠ 0
Cofactors of all elements of matrix A are
A11 = `(-1)^2 |("( )", "( )"),("( )", "( )")|` = (______),
A12 = `(-1)^3 |(5, "( )"),("( )", 3)|` = – 15,
A13 = `(-1)^4 |(5, "( )"),("( )", 1)|` = 5,
A21 = _______, A22 = _______, A23 = _______,
A31 = `(-1)^4 |("( )", "( )"),("( )", "( )")|` = (______),
A32 = `(-1)^5 |(2, "( )"),("( )", 0)|` = ( ),
A33 = `(-1)^6 |(2, "( )"),("( )", 1)|` = 2,.
Cofactors of matrix A = `[(3, "____", "____"),("____", "____",-2),(1, "____", "____")]`
adj (A) = `[("____", "____", "____"),("____", "____","____"),("____","____","____")]`
A.adj (A) = `[(2, 0, -1),(5, 1, 0),(0, 1, 3)] [("( )", -1, 1), (-15, "( )", -5),("( )", -2, "( )")] = [(1, 0, "( )"),("( )", "( )", "( )"),(0, "( )", "( )")]` = |A|I
If A = `[(1,3,3),(1,4,3),(1,3,4)]` then verify that A(adj A) = |A| I and also find A-1.
Find the inverse of the following matrix:
`[(1,-1),(2,3)]`
Find the inverse of the following matrix:
`[(3,1),(-1,3)]`
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.
The inverse matrix of `((4/5,(-5)/12),((-2)/5,1/2))` is
If A and B non-singular matrix then, which of the following is incorrect?
If A = `|(3,-1,1),(-15,6,-5),(5,-2,2)|` then, find the Inverse of A.
If A is non-singular matrix and (A + l)(A - l) = 0 then A + A-1 = ______.
The inverse of `[(1,cos alpha),(- cos alpha, -1)]` is ______.
If A2 - A + I = 0, then A-1 = ______.
If A = `[(1, tanx),(-tanx, 1)]`, then AT A–1 = ______.
If A = `[(5, -4), (7, -5)]`, then 3A-1 = ______
If `A = [[-3,1],[-4,3]]` and A-1 = αA, then α = ______.
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 ______.
If A = `[(cos theta, sin theta, 0),(-sintheta, costheta, 0),(0, 0, 1)]`, where A11, A11, A13 are co-factors of a11, a12, a13 respectively, then the value of a11A11 + a12A12 + a13A13 = ______.
If A–1 = `[(3, -1, 1),(-15, 6, -5),(5, -2, 2)]`, then adj A = ______.
If A = `[(-i, 0),(0, i)]`, then ATA is equal to
If A = `[(2, 3),(a, 6)]` is a singular matrix, then a = ______.
If A = `[(1, 2, -1),(-1, 1, 2),(2, -1, 1)]`, then det (adj (adj A)) is ______.
If A = `[(1, 2, 3),(-1, 1, 2),(1, 2, 4)]` then (A2 – 5A)A–1 = ______.
If A = `[(2, 3),(4, 5)]`, show that A2 – 7A – 2I = 0
If A = `[(1, 2, 4),(4, 3, -2),(1, 0, -3)]`. Show that A–1 exists and find A–1 using column transformation.
