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Express A^-1 in terms of A using a quadratic equation A^2 + pA + qI = O

This type of question, asked 11 times: 4 May 2023 S1, 4 May 2023 S2, 16 May 2023 S2, 12 May 2024 S1, 19 Apr 2025 S2, 26 Apr 2025 S1, 26 Apr 2025 S2, 17 Apr 2026 S1, 12 May 2026 S2, 14 May 2026 S1, 19 May 2026 S1. All 11 questions below, oldest first.

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  1. Question 1

    4 May 2023 S1

    101
    Suppose A is any 3×33\times3 non-singular matrix and (A−3I)(A−5I)=O(A-3I)(A-5I)=O where I=I3I=I_3 and O=O3O=O_3. Here O3O_3 represent zero matrix of order 3 and I3I_3 is an identity matrix of order 3. If αA+βA−1=4I\alpha A + \beta A^{-1} = 4I, then α+β\alpha + \beta is equal to

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  2. Question 2

    4 May 2023 S2

    129
    Let A=(12−51)A=\begin{pmatrix}1&2\\-5&1\end{pmatrix} and A−1=xA+yI2A^{-1}=xA+yI_2, (where I2I_2 is unit matrix of order 2), then

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  3. Question 3

    16 May 2023 S2

    115
    Let A=(12−14)A = \begin{pmatrix}1 & 2 \\ -1 & 4\end{pmatrix} and A−1=αI+βAA^{-1} = \alpha I + \beta A, α,β∈R\alpha, \beta \in \mathbb{R}, II is the identity matrix of order 2, then 4(α−β)4(\alpha - \beta) is

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  4. Question 4

    12 May 2024 S1

    139
    If the matrix A=(12−51)A=\begin{pmatrix}1&2\\-5&1\end{pmatrix} and A−1=xA+yIA^{-1}=xA+yI when II is a unit matrix of order 2, then the value of 2x+3y2x+3y is

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  5. Question 5

    19 Apr 2025 S2

    127
    If A=[12−14]A=\begin{bmatrix} 1 & 2 \\ - 1 & 4 \end{bmatrix} and A−1=αI+βAα,β∈RA^{- 1}=\alpha I+\beta A\alpha,\beta\in R where II is the identity matrix of order 2 , hen 4(α+β)=4(\alpha+\beta) =

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  6. Question 6

    26 Apr 2025 S1

    143
    Matrix A is non-singular matrix and (A−3I)(A−5I)=0(A- 3I)(A- 5I) = 0, then 158A−1=\frac{15}{8}A^{- 1}= ____\_\_\_\_

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  7. Question 7

    26 Apr 2025 S2

    111
    If A is a matrix of order 2 and I is the identity matrix of order 2 such that A2−4 A+3I=0A^{2}- 4\text{ }A + 3I = 0 then (A+3I)−1=(A + 3I)^{- 1}=

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  8. Question 8

    17 Apr 2026 S1

    66
    If AA is a non-singular matrix and A2−A+I=0A^2 - A + I = 0, then A−1=…A^{-1} = \ldots

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  9. Question 9

    12 May 2026 S2

    67
    If A=[122212221]A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix} and A−1=xA+yIA^{-1} = xA + yI, then the values of xx and yy are...

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  10. Question 10

    14 May 2026 S1

    68
    If A=[4521]A = \begin{bmatrix} 4 & 5 \\ 2 & 1 \end{bmatrix}, then A−1=A^{-1} =

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  11. Question 11

    19 May 2026 S1

    69
    Let A=[12−51]A = \begin{bmatrix} 1 & 2 \\ -5 & 1 \end{bmatrix} and A−1=xA+yIA^{-1} = xA + yI, then the pair (x,y)=⋯(x, y) = \cdots

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