Advertisements
Advertisements
प्रश्न
If A = `[(1,-1),(2,3)]` show that A2 - 4A + 5I2 = 0 and also find A-1.
उत्तर
Given A = `[(1,-1),(2,3)] |"A"| = |(1,-1),(2,3)|` = 3 + 2 = 5 ≠ 0
∴ A-1 exists.
LHS = A2 - 4A + 5I
Premultiply by A-1 we get
LHS = `"A"^-1. "A"^2 - 4"A"^-1."A" + 5"A"^-1."I"`
`= ("A"^-1 "A")."A" - 4"I" + 5"A"^-1` ...[∵ A-1A = I]
= IA - 4I + 5A-1
= A - 4I + 5A-1
Now `"A"^-1 = 1/|"A"| "adj A"`
`= 1/5 [(3,1),(-2,1)]`
∴ LHS = `[(1,-1),(2,3)] - 4[(+1,0),(0,1)] + 5 . 1/5 [(3,1),(-2,1)]`
`= [(1,-1),(2,3)] + [(-4,0),(0,-4)] + [(3,1),(-2,1)]`
`= [(1-4+3,-1+0+1),(2+0-2,3-4+1)] = [(0,0),(0,0)]`
= RHS
APPEARS IN
संबंधित प्रश्न
Find the inverse of the following matrices by the adjoint method `[(1, 2, 3),(0, 2, 4),(0, 0, 5)]`.
Fill in the blank :
If A = `[(3, -5),(2, 5)]`, then co-factor of a12 is _______
Check whether the following matrices are invertible or not:
`[(1, 2, 3),(2, 4, 5),(2, 4, 6)]`
Find inverse of the following matrices (if they exist) by elementary transformations :
`[(1, -1),(2, 3)]`
If A = `[("a", "b"),("c", "d")]` then find the value of |A|−1
If A = `[(1,3,3),(1,4,3),(1,3,4)]` then verify that A(adj A) = |A| I and also find A-1.
If A is 3 × 3 matrix and |A| = 4 then |A-1| is equal to:
The matrix A = `[("a",-1,4),(-3,0,1),(-1,1,2)]` is not invertible only if a = _______.
If A is non-singular matrix such that (A - 2l)(A - 4l) = 0 then A + 8A-1 = ______.
If matrix A = `[(3, -2, 4),(1, 2, -1),(0, 1, 1)]` and A–1 = `1/k` (adj A), then k is ______.
