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Math Help - Matrix

  1. #1
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    Matrix

    1.) If A is \begin{bmatrix} 3 & 2 \\-2 & -1 \end{bmatrix}, write A^2 in the form pA+qI where p and q are scalars. Hence write A^(-1) in the form rA+sI where r and s are scalars.

    I know how to find A^2, I got \begin{bmatrix} 5 & 4 \\-4 & -3 \end{bmatrix} but I do not know how to convert this matrix form into linear form pA+qI

    2.) It is known that AB=A and BA=B where matrices A and B are not necessarily invertible.
    Prove that A^2 = A.

    When I first saw this, I thought B had to be I in AB=A and A in BA=B had to be I BUT they then added, NOTE: From AB=A, you cannot deduce that B=I. They asked me why, and I really dont know since I thought you could deduce that!
    How do I then prove that A^2=A?
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  2. #2
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    Re: Matrix

    Quote Originally Posted by Tutu View Post
    1.) If A is \begin{bmatrix} 3 & 2 \\-2 & -1 \end{bmatrix}, write A^2 in the form pA+qI where p and q are scalars. Hence write A^(-1) in the form rA+sI where r and s are scalars.

    I know how to find A^2, I got \begin{bmatrix} 5 & 4 \\-4 & -3 \end{bmatrix} but I do not know how to convert this matrix form into linear form pA+qI

    2.) It is known that AB=A and BA=B where matrices A and B are not necessarily invertible.
    Prove that A^2 = A.

    When I first saw this, I thought B had to be I in AB=A and A in BA=B had to be I BUT they then added, NOTE: From AB=A, you cannot deduce that B=I. They asked me why, and I really dont know since I thought you could deduce that!
    How do I then prove that A^2=A?
    \displaystyle \begin{align*} p\mathbf{A} + q\mathbf{I} &= p\left[ \begin{matrix} \phantom{-}5 & \phantom{-}4 \\ -4 & -3 \end{matrix} \right] + q \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right] \\ &= \left[ \begin{matrix} \phantom{-}5p & \phantom{-}4p \\ -4p & -3p \end{matrix} \right] + \left[ \begin{matrix} q & 0 \\ 0 & q \end{matrix} \right] \\ &= \left[ \begin{matrix} \phantom{-}5p + q & \phantom{-}4p \\ -4p & -3p + q \end{matrix} \right] \end{align*}

    If this is equal to \displaystyle \mathbf{A}^2 , then that means you can set each of the components equal and solve for p and q.
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  3. #3
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    Re: Matrix

    I see thank you so so much! ((:
    Any ideas for the second question?

    Thanks!
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    Re: Matrix

    Quote Originally Posted by Tutu View Post
    1.) If A is \begin{bmatrix} 3 & 2 \\-2 & -1 \end{bmatrix}, write A^2 in the form pA+qI where p and q are scalars. Hence write A^(-1) in the form rA+sI where r and s are scalars.

    I know how to find A^2, I got \begin{bmatrix} 5 & 4 \\-4 & -3 \end{bmatrix} but I do not know how to convert this matrix form into linear form pA+qI

    2.) It is known that AB=A and BA=B where matrices A and B are not necessarily invertible.
    Prove that A^2 = A.

    When I first saw this, I thought B had to be I in AB=A and A in BA=B had to be I BUT they then added, NOTE: From AB=A, you cannot deduce that B=I. They asked me why, and I really dont know since I thought you could deduce that!
    How do I then prove that A^2=A?
    \displaystyle \mathbf{A}\mathbf{B} = \mathbf{A} and \displaystyle \mathbf{B}\mathbf{A} = \mathbf{B}. Then

    \displaystyle \begin{align*} \mathbf{A}^2 &= \left( \mathbf{A}\mathbf{B} \right)^2 \\ &= \mathbf{A}\mathbf{B}\mathbf{A}\mathbf{B} \\ &= \mathbf{A}\mathbf{B}\mathbf{B} \\ &= \mathbf{A}\mathbf{B} \\ &= \mathbf{A} \end{align*}
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