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Math Help - Matrix question.

  1. #1
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    Matrix question.

    If A is a 2X2 matrix, a12 = a21 = 0, a11 ≠ 0, a22 ≠ 0 then show that A is regular and find A^-1.
    Many thanks in advance.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by grooverandshaker View Post
    If A is a 2X2 matrix, a12 = a21 = 0, a11 ≠ 0, a22 ≠ 0 then show that A is regular and find A^-1.
    Many thanks in advance.
    A matrix is regular if it has an inverse. How can you tell if a matrix has an inverse? (Hint: Think of what the determinant of a singular matrix is.)

    -Dan
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  3. #3
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    Um, I know that a matrix has an inverse if the determinant is nonzero. Is that right?
    I usually know how to find the determinant and the inverse but I just can't do it when the matrix isn't written out.
    Could you help me figure out what the original matrix is?
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  4. #4
    Member rtblue's Avatar
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    Given that a12 and a21 are both 0 and that a11 and a22 are not 0, the matrix must have a nonzero determinant. There are no values of a11 and a22 that can make the determinant 0.
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by grooverandshaker View Post
    Um, I know that a matrix has an inverse if the determinant is nonzero. Is that right?
    I usually know how to find the determinant and the inverse but I just can't do it when the matrix isn't written out.
    Could you help me figure out what the original matrix is?

    This page
    seems to be a good guide in general. It has an explicit answer for the 2 x 2 case, but make sure you understand how to do the problem in general.

    -Dan
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  6. #6
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    \displaystyle A=\begin{bmatrix}\alpha & 0\\0&\beta\end{bmatrix} \ \ \ \ \alpha, \ \beta\neq 0

    \displaystyle\text{det}(A)=\alpha\cdot\beta-0\neq 0

    \displaystyle A^{-1}=\frac{1}{\text{det}(A)}A
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  7. #7
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    Thanks guys. I think I understand now.
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