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Value of a matrix polynomial such as A^2 + pA + qI

This type of question, asked 3 times: May 2021 S1, 16 Apr 2026 S2, 18 Apr 2026 S2. All 3 questions below, oldest first.

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  1. Question 1

    May 2021 S1

    117
    If matrix A=(1−123)A = \begin{pmatrix}1&-1\\2&3\end{pmatrix}, then A2−4A+5IA^2 - 4A + 5I is where II is a unit matrix.

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  2. Question 2

    16 Apr 2026 S2

    65
    If A=[2−102]A = \begin{bmatrix} 2 & -1 \\ 0 & 2 \end{bmatrix} and A2+xA+yI2=O2A^2 + xA + yI_2 = O_2, where I2I_2 and O2O_2 are the identity matrix and null matrix of order 2 respectively, then:

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  3. Question 3

    18 Apr 2026 S2

    64
    Let A=[−3214]A = \begin{bmatrix} -3 & 2 \\ 1 & 4 \end{bmatrix} and if A2−2A+I=[18pq11]A^2 - 2A + I = \begin{bmatrix} 18 & p \\ q & 11 \end{bmatrix}, then …\ldots

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