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Math Help - Matrices!

  1. #1
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    Matrices!

    How do I do

    If A = \begin{bmatrix} 3 & 2 \\-2 & -1 \end{bmatrix}, write A^2 in the form pA+qI where p and q are scalars?

    How do I convert a square matrix to _A+_I?

    Really need help, thank you so so much!
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  2. #2
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    Re: Matrices!

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  3. #3
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    Re: Matrices!

    I see no reason not to just do this by straightforward calculation.

    Since the problem asks about A^2, the obvious first thing to do is to square A!
    \begin{bmatrix}3 & 2 \\ -2 & -1\end{bmatrix}\begin{bmatrix}3 & 2 \\ -2 & -1\end{bmatrix}= \begin{bmatrix}5 & 4 \\ -4 & -3 \end{bmatrix}

    I presume you know that I is the identity matrix: \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} so that, for any q, qI= \begin{bmatrix}q & 0 \\ 0 & q\end{bmatrix} and then A^2- qI= \begin{bmatrix}5- q & 4 \\ -4 & -3- q\end{bmatrix} and that must be equal to pA= \begin{bmatrix}3p & 2p \\ -2p & -p\end{bmatrix}.

    That is, we have the four equations, 5- q= 3p, 4= 2p, -4= -2p, and -3- q= -p. Solve those equations for p and q.
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