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Matrices and Determinants question

2020 · Shift 2 · Q22
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  5. /2020 · Shift 2 · Q22

Matrices and Determinants question

2020 · Shift 2 · Q22

JEE AdvancedMathematicsMatrices and DeterminantsNumerical+3 / −1
The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2 ×\times× 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 .............
Numerical answer
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Correct answer: 5

  1. Let the eigenvalues of the 2×22\times 22×2 matrix AAA be α\alphaα and β\betaβ.

    Then: α+β=tr⁡(A)=3\alpha+\beta=\operatorname{tr}(A)=3α+β=tr(A)=3 and α3+β3=tr⁡(A3)=−18.\alpha^3+\beta^3=\operatorname{tr}(A^3)=-18.α3+β3=tr(A3)=−18.

  2. We use the identity: α3+β3=(α+β)3−3αβ(α+β).\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta).α3+β3=(α+β)3−3αβ(α+β).

    Substituting α+β=3\alpha+\beta=3α+β=3: α3+β3=33−3αβ⋅3=27−9αβ.\alpha^3+\beta^3=3^3-3\alpha\beta\cdot 3=27-9\alpha\beta.α3+β3=33−3αβ⋅3=27−9αβ.

  3. Given that α3+β3=−18\alpha^3+\beta^3=-18α3+β3=−18, we get: 27−9αβ=−18.27-9\alpha\beta=-18.27−9αβ=−18.

    So, −9αβ=−45-9\alpha\beta=-45−9αβ=−45 αβ=5.\alpha\beta=5.αβ=5.

  4. For a 2×22\times 22×2 matrix, the determinant equals the product of eigenvalues: det⁡(A)=αβ=5.\det(A)=\alpha\beta=5.det(A)=αβ=5.

  5. Therefore, the determinant of AAA is 5.\boxed{5}.5​.

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