Advertisements
Advertisements
Question
Find the inverse of matrix B = `[(3,1, 5),(2, 7, 8),(1, 2, 5)]` by using adjoint method
Solution
B = `[(3,1, 5),(2, 7, 8),(1, 2, 5)]`
∴ |B| = `|(3, 1, 5),(2, 7, 8),(1, 2, 5)|`
= 3(35 – 16) – 1(10 – 8) + 5(4 – 7)
= 3(19) – 1(2) + 5(– 3)
= 57 – 2 – 15
= 40 ≠ 0
∴ B–1 exists.
Here,
b11 = 3
∴ M11 = `|(7, 8),(2, 5)|`
= 35 – 16
= 19
and B11 = (–1)1+1 M11 = 19
b12 = 1
∴ M12 = `|(2, 8),(1, 5)|`
= 10 – 8
= 2
and B12 = (–1)1+2 M12 = – 2
b13 = 5
∴ M13 = `|(2, 7),(1, 2)|`
= 4 – 7
= – 3
and B13 = (–1)1+3 M13 = – 3
b21 = 2
∴ M21 = `|(1, 5),(2, 5)|`
= 5 – 10
= – 5
and B21 = (–1)2+1 M21 = 5
b22 = 7
∴ M22 = `|(3, 5),(1, 5)|`
= 15 – 5
= 10
and B22 = ( –1)2+2 M22 = 10
b23 = 8
∴ M23 = `|(3, 1),(1, 2)|`
= 6 – 1
= 5
and B23 = ( –1)2+3 M23 = – 5
b31 = 1
∴ M31 = `|(1, 5),(7, 8)|`
= 8 – 35
= – 27
and B31 = ( –1)3+1 M31 = – 27
b32 = 2
∴ M32 = `|(3, 5),(2, 8)|`
= 24 – 10
= 14
and B32 = ( –1)3+2 M32 = – 14
b33 = 5
∴ M33 = `|(3, 1),(2, 7)|`
= 21 – 2
= 19
and B33 = ( –1)3+3 M33 = 19
∴ The matrix of the co-factors is
[Bij]3×3 = `[("B"_11, "B"_12, "B"_13),("B"_21, "B"_22, "B"_23),("B"_31, "B"_32, "B"_33)]`
= `[(19, -2, -3),(5, 10, -5),(-27, -4, 19)]`
Now, adj B = `["B"_"ij"]_(3 xx 3)^"T"`
= `[(19, 5, -27),(-2, 10, -14),(-3, -5, 9)]`
∴ B–1 = `1/|"B"|` (adj B)
= `1/40 [(19, 5, -27),(-2, 10, -14),(-3, -5, 9)]`
APPEARS IN
RELATED QUESTIONS
Find the co-factor of the element of the following matrix.
`[(1,-1,2),(-2,3,5),(-2,0,-1)]`
Find the adjoint of the following matrix.
`[(2,-3),(3,5)]`
Find the inverse of the following matrix.
`[(1,2),(2,-1)]`
Find the inverse of the following matrix (if they exist):
`[(2,1),(7,4)]`
Find the inverse of `[(1,2,3),(1,1,5),(2,4,7)]` by the adjoint method.
Choose the correct answer from the given alternatives in the following question:
If A = `[(2,-4),(3,1)]`, then the adjoint of matrix A is
Find the inverse of the following matrices by the adjoint method `[(3, -1),(2, -1)]`.
Fill in the blank :
If A = [aij]mxm is a non-singular matrix, then A–1 = `(1)/(......)` adj(A).
State whether the following is True or False :
If A and B are conformable for the product AB, then (AB)T = ATBT.
If A = `[(3, 0, 0),(0, 3, 0),(0, 0, 3)]`, then |A| |adj A| = ______
If the inverse of the matrix `[(alpha, 14, -1),(2, 3, 1),(6, 2, 3)]` does not exists then find the value of α
Find A–1 using adjoint method, where A = `[(cos theta, sin theta),(-sin theta, cos theta)]`
Find the adjoint of matrix A = `[(6, 5),(3, 4)]`
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 Cofactor of element a21 in matrix A = `[(1, 2),(5, -8)]` is ______
If A = `[(1,3,3),(1,4,3),(1,3,4)]` then verify that A(adj A) = |A| I and also find A-1.
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.
Solve by matrix inversion method:
x – y + 2z = 3; 2x + z = 1; 3x + 2y + z = 4
Weekly expenditure in an office for three weeks is given as follows. Assuming that the salary in all the three weeks of different categories of staff did not vary, calculate the salary for each type of staff, using the matrix inversion method.
| Week | Number of employees | Total weekly salary (in ₹) |
||
| A | B | C | ||
| 1st week | 4 | 2 | 3 | 4900 |
| 2nd week | 3 | 3 | 2 | 4500 |
| 3rd week | 4 | 3 | 4 | 5800 |
adj (AB) is equal to:
If A is 3 × 3 matrix and |A| = 4 then |A-1| is equal to:
Solve by using matrix inversion method:
x - y + z = 2, 2x - y = 0, 2y - z = 1
If A is non-singular matrix and (A + l)(A - l) = 0 then A + A-1 = ______.
If ω is a complex cube root of unity and A = `[(ω,0,0),(0,ω^2,0),(0,0,1)]` then A-1 = ?
If A = `[(2, -3), (3, 5)]`, then |Adj A| is equal to ______
If A = `[(5, -4), (7, -5)]`, then 3A-1 = ______
If A = `[(1,-1,1),(2,1,-3),(1,1,1)]`, then the sum of the elements of A-1 is ______.
If `A = [[-3,1],[-4,3]]` and A-1 = αA, then α = ______.
If A, B are two square matries, such that AB = B, BA = A and n ∈ N then (A + B)n =
If matrix P = `[(0, -tan (θ//2)),(tanθ//2, 0)]`, then find (I – P) `[(cosθ, -sinθ),(sinθ, cosθ)]`
If A = `[(0, 0, 1),(0, 1, 0),(1, 0, 0)]`, then A2008 is equal to ______.
If A = `[(x, 1),(1, 0)]` and A = A–1, then x = ______.
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 ______.
For an invertible matrix A, if A (adj A) = `|(20, 0),(0, 20)|`, then | A | = ______.
If matrix A = `[(1, -1),(2, 3)]`, then A2 – 4A + 5I is where I is a unit matix.
If A = `[(2, 3),(4, 5)]`, show that A2 – 7A – 2I = 0
Find the inverse of the matrix `[(1, 1, 1),(1, 2, 3),(3, 2, 2)]` by elementary column transformation.
