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The trace and determinant of a 2 × 2 matrix are known to be –2 and –35 respectively. Its eigenvalues are :
- a)–30 and –5
- b)–37 and –1
- c)–7 and 5
- d)17.5 and –2
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The trace and determinant of a 2 × 2 matrix are known to be –2 and –3...
∴ Trace = λ1 + λ2 = -2
Determinant = λ1 λ2 = -35
Solving , we get λ1 = -7, and λ2 = 5
∴ a + d = -2
and ad – bc = -35
Alternately
Now,
= (λ – a) (λ – d) – bc = 0
λ2 – (a + d) λ – bc + ad = 0
λ2 + 2λ – 35 = 0
⇒ (λ + 7) (λ – 5) = 0
λ = -7 or λ = 5
Most Upvoted Answer
The trace and determinant of a 2 × 2 matrix are known to be –2 and –3...
The trace of a matrix is the sum of its diagonal elements, and the determinant of a 2 × 2 matrix can be found by subtracting the product of the two diagonal elements from the product of the off-diagonal elements.
Given that the trace of the matrix is –2 and the determinant is –35, we can set up the following equations:
Equation 1: a + d = –2 (where a and d are the diagonal elements of the matrix)
Equation 2: ad – bc = –35 (where a, b, c, and d are the elements of the 2 × 2 matrix)
Solving these equations, we can find the values of a, b, c, and d.
Solving Equation 1:
We can rewrite Equation 1 as:
d = –2 – a
Substituting in Equation 2:
ad – bc = –35
a(–2 – a) – bc = –35
–2a – a^2 – bc = –35
a^2 + 2a + bc = 35
Substituting d = –2 – a:
a^2 + 2a + b(–2 – a) = 35
a^2 + 2a – 2b – ab = 35
a^2 + (2 – b)a – 2b = 35
Using the values of trace and determinant:
a^2 + (2 – b)a – 2b = 35
a^2 + (2 – b)a – 2b + 35 = 0
a^2 + (2 – b)a – 2(b – 17.5) = 0
Factoring the equation:
(a – 5)(a + 7) = 0
Hence, the eigenvalues are a = 5 and a = –7.
Therefore, the correct answer is option C: –7 and 5.
Given that the trace of the matrix is –2 and the determinant is –35, we can set up the following equations:
Equation 1: a + d = –2 (where a and d are the diagonal elements of the matrix)
Equation 2: ad – bc = –35 (where a, b, c, and d are the elements of the 2 × 2 matrix)
Solving these equations, we can find the values of a, b, c, and d.
Solving Equation 1:
We can rewrite Equation 1 as:
d = –2 – a
Substituting in Equation 2:
ad – bc = –35
a(–2 – a) – bc = –35
–2a – a^2 – bc = –35
a^2 + 2a + bc = 35
Substituting d = –2 – a:
a^2 + 2a + b(–2 – a) = 35
a^2 + 2a – 2b – ab = 35
a^2 + (2 – b)a – 2b = 35
Using the values of trace and determinant:
a^2 + (2 – b)a – 2b = 35
a^2 + (2 – b)a – 2b + 35 = 0
a^2 + (2 – b)a – 2(b – 17.5) = 0
Factoring the equation:
(a – 5)(a + 7) = 0
Hence, the eigenvalues are a = 5 and a = –7.
Therefore, the correct answer is option C: –7 and 5.
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The trace and determinant of a 2 × 2 matrix are known to be –2 and –35 respectively. Its eigenvalues are :a)–30 and –5b)–37 and –1c)–7 and 5d)17.5 and –2Correct answer is option 'C'. Can you explain this answer?
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