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12.4 Solutions to the Systems of Equations Problems

  1. No, we do not know that A has an inverse.

  2. I. This is because AA-1 = I, A-1A = I, IT = I and II = I.

  3. On these problems, we have provided the solution and one of the paths that leads to that solution. There are different paths, but only one solution.

    1.   matrix

      matrix


    2.   columm vector x

      augmented matrix


    3.   column vector x

      augmented matrix


    4.   column vector x

      augmented matrix



    1.   A inverse Use the formula that we found for 2 by 2 matrices.

    2.   A inverse Use the formula that we found for 2 by 2 matrices.

    3.   A inverse


    1. Live

    2. Love

    3. Learn

    4. Answers will vary

  6. Ax = b. Therefore, if A-1 exists (which it does for the stated problems), AA-1x = A-1b. Therefore, x = A-1b.

    1.   solution for x

    2.   solution for x

    3.   solution for x

    4.   solution for x

    5.   solution for x

    6.   solution for x

    7.   solution for x

  7. A = ( A-1)-1 since AA-1 = I and A-1A = I, A and A-1 are inverses of each other. Therefore, the inverse of A-1 is A.

    1.   inverse matrix

    2. If A is a diagonal matrix consisting of the elements aii, then A-1 is a diagonal matrix consisting of the elements 1/a_ii

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Updated: September 23, 2000

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