Let I be an identity matrix of order 2x2 and P = [[2, -1], [5, -3]]. Then the value of n ∈ N for which Pⁿ = 5I - 8P is equal to......
Let I be an identity matrix of order 2x2 and P = [[2, -1], [5, -3]]. Then the value of n ∈ N for which Pⁿ = 5I - 8P is equal to......
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1 Answer
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Given the matrix P = [2, -1], [5, -3].
The characteristic equation is det (P - λI) = 0, which is (2-λ) (-3-λ) - (-1) (5) = 0.
This simplifies to λ² + λ - 1 = 0.
By the Cayley-Hamilton theorem, the matrix P satisfies this equation: P² + P - I = 0, so P² = I - P.
To find P³: P³ = P * P² = P (I-P) = P - P² = P - (I-P) = 2P - I.
The problem asks for N=6, likely related to a higher power P? Continuing the pattern:
P? = 2P² - P = 2 (I-P) - P = 2I - 3P.
P? = 2P - 3P² = 2P - 3 (I-P) = 5P - 3I.
P? = 5P² - 3P = 5 (I-P) - 3P = 5I - 8P.
The solution N=6 must relate to a di...more
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Let
Given ...(1)
∴ x1 + z1 = 2 … (2)
x2 + z2 = 0 … (3)
x3 + z3 = 0 … (4)
Given
⇒ – x1 + z1 = −4 … (5)
–x2 + z2 = 0 &nbs
g (x) = px + q
Compare 8 = ap2 …………… (i)
-2 = a (2pq) + bp
0 = aq2 + bq + c
=>4x2 + 6x + 1 = apx2 + bpx + cp + q
=> Andhra Pradesh = 4 ……………. (ii)
6 = bp
1 = cp + q
From (i) & (ii), p = 2, q = -1
=> b = 3, c = 1, a = 2
f (x) = 2x2 + 3x + 1
f (2) = 8 + 6 + 1 = 15
g (x) = 2x – 1
g (2) = 3
Kindly consider the following figure
B = (I – adjA)5
Kindly consider the following figure
B = (I – adjA)5
System of equation is
R1 – 2 R2, R3 – R2
System of equation will have no solution for = -7.
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