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The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 × 2 matrix such that the trace of A is 3 and the trace of A3 is −18, then the value of the determinant of A is _____
Correct answer is '5'. Can you explain this answer?
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The trace of a square matrix is defined to be the sum of its diagonal ...
Given trace(A) = a + d = 3
and trace(A3) = a3 + d3 + 3abc + 3bcd = –18
⇒ a3 + d3 + 3bc(a + d) = -18
⇒ a3 + d3 + 9bc = -18
⇒ (a + d)((a + d)2 - 3ad) + 9bc = -18
⇒ 3(9 - 3ad) + 9bc = -18
⇒ ad - bc = 5 = determinant of A
|A – λI| = (a - λ)(d - λ) - bc
= λ2 - (a + d)λ + ad - bc
= λ2 - 3λ + Δ
⇒ O = A2 - 3A + ΔI
⇒ A2 = 3A - ΔI
⇒ A3 = 3A2 - ΔA
= 3(3A – ΔI) - ΔA
= (9 – Δ)A - 3ΔI
∴ trace A3 = (9 - Δ)(a + d) - 6Δ
⇒ -18 = (9 - Δ)(3) - 6Δ
= 27 – 9Δ
⇒ 9Δ = 45 ⇒ Δ = 5
Most Upvoted Answer
The trace of a square matrix is defined to be the sum of its diagonal ...
2 matrix with trace 5 and determinant 2, find the eigenvalues of A.
By the properties of trace and determinant, we know that the sum of the eigenvalues of A is 5 and the product of the eigenvalues is 2. Let λ1 and λ2 be the eigenvalues of A.
Then, we have:
λ1 + λ2 = 5
λ1λ2 = 2
Solving for λ1 and λ2, we can use the quadratic formula:
λ1,2 = (5 ± √(5^2 - 4(2)))/2
λ1,2 = (5 ± √17)/2
Therefore, the eigenvalues of A are (5 + √17)/2 and (5 - √17)/2.
By the properties of trace and determinant, we know that the sum of the eigenvalues of A is 5 and the product of the eigenvalues is 2. Let λ1 and λ2 be the eigenvalues of A.
Then, we have:
λ1 + λ2 = 5
λ1λ2 = 2
Solving for λ1 and λ2, we can use the quadratic formula:
λ1,2 = (5 ± √(5^2 - 4(2)))/2
λ1,2 = (5 ± √17)/2
Therefore, the eigenvalues of A are (5 + √17)/2 and (5 - √17)/2.
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The trace of a square matrix is defined to be the sum of its diagonal entries. If Ais a 2 × 2 matrix such that the trace of Ais 3 and the trace of A3 is −18, then the value of the determinant of Ais _____Correct answer is '5'. Can you explain this answer?
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The trace of a square matrix is defined to be the sum of its diagonal entries. If Ais a 2 × 2 matrix such that the trace of Ais 3 and the trace of A3 is −18, then the value of the determinant of Ais _____Correct answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The trace of a square matrix is defined to be the sum of its diagonal entries. If Ais a 2 × 2 matrix such that the trace of Ais 3 and the trace of A3 is −18, then the value of the determinant of Ais _____Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The trace of a square matrix is defined to be the sum of its diagonal entries. If Ais a 2 × 2 matrix such that the trace of Ais 3 and the trace of A3 is −18, then the value of the determinant of Ais _____Correct answer is '5'. Can you explain this answer?.
The trace of a square matrix is defined to be the sum of its diagonal entries. If Ais a 2 × 2 matrix such that the trace of Ais 3 and the trace of A3 is −18, then the value of the determinant of Ais _____Correct answer is '5'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The trace of a square matrix is defined to be the sum of its diagonal entries. If Ais a 2 × 2 matrix such that the trace of Ais 3 and the trace of A3 is −18, then the value of the determinant of Ais _____Correct answer is '5'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The trace of a square matrix is defined to be the sum of its diagonal entries. If Ais a 2 × 2 matrix such that the trace of Ais 3 and the trace of A3 is −18, then the value of the determinant of Ais _____Correct answer is '5'. Can you explain this answer?.
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