Diagonalisation of symmetric matrices

Diagonalisation of symmetric matrices Quiz

Spec 6.8 · Edexcel International A Level Further Maths: Quiz

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5 questions, one answer each

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  1. 1. Which matrix is orthogonal?

    • A. (11−11)\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}
    • B. (1221)\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}
    • C. (0.60.8−0.80.6)\begin{pmatrix} 0.6 & 0.8 \\ -0.8 & 0.6 \end{pmatrix}
    • D. (0.60.80.80.6)\begin{pmatrix} 0.6 & 0.8 \\ 0.8 & 0.6 \end{pmatrix}

  2. 2. A symmetric matrix has distinct eigenvalues and one eigenvector (12)\begin{pmatrix} 1 \\ 2 \end{pmatrix}. Which could be an eigenvector for another eigenvalue?

    • A. (2−1)\begin{pmatrix} 2 \\ -1 \end{pmatrix}
    • B. (21)\begin{pmatrix} 2 \\ 1 \end{pmatrix}
    • C. (−1−2)\begin{pmatrix} -1 \\ -2 \end{pmatrix}
    • D. (1−2)\begin{pmatrix} 1 \\ -2 \end{pmatrix}

  3. 3. For a symmetric A\mathbf{A}, the first column of P\mathbf{P} is for λ=7\lambda=7 and the second for λ=3\lambda=3. What is D\mathbf{D}?

    • A. (3007)\begin{pmatrix} 3 & 0 \\ 0 & 7 \end{pmatrix}
    • B. (7337)\begin{pmatrix} 7 & 3 \\ 3 & 7 \end{pmatrix}
    • C. (7007)\begin{pmatrix} 7 & 0 \\ 0 & 7 \end{pmatrix}
    • D. (7003)\begin{pmatrix} 7 & 0 \\ 0 & 3 \end{pmatrix}

  4. 4. P\mathbf{P} is orthogonal and PTAP=diag(1,6)\mathbf{P}^T\mathbf{A}\mathbf{P}=\mathrm{diag}(1,6). What is det⁡A\det\mathbf{A}?

    • A. 77
    • B. 66
    • C. 55
    • D. 11

  5. 5. S=(1221)\mathbf{S}=\begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix} has eigenvalues 33 and −1-1. What is a normalised eigenvector for −1-1?

    • A. 12(11)\frac{1}{\sqrt2}\begin{pmatrix} 1 \\ 1 \end{pmatrix}
    • B. 12(1−1)\frac{1}{2}\begin{pmatrix} 1 \\ -1 \end{pmatrix}
    • C. 12(1−1)\frac{1}{\sqrt2}\begin{pmatrix} 1 \\ -1 \end{pmatrix}
    • D. (1−1)\begin{pmatrix} 1 \\ -1 \end{pmatrix}

Quiz