Essential Mathematics: Matrices

7 flashcards ยท Mathematics ยท by lakesare

  1. Suppose there's a network of water pipes. How to represent it as a matrix? Water can be poured into the system at position A or B, and it then flows down the pipes to come out at positions U, V and W. The pipes have different capacities: the three pipes that come out of position A are labelled with numbers that indicate the proportions of water that flow through these pipes when water is poured in at A, and the three pipes that come out of position B are labelled in the same way.

    Here the elements in each column are the labels of the arcs from a node (A or B), and the elements in each row are the labels of the arcs to a node (U, V or W).

  2. [About equivalent networks] The network below is the combination of two smaller networks, one with input nodes U and V and output nodes P and Q, and the other with input nodes P and Q and output nodes A, B, and C.

  3. What's the inverse of

  4. When is the 2x2 matrix invertible?

    Matrix is invertible: when determinant (ad - bc) isn't zero.

  5. How are a system of linear equations, and determinant of matrix of its coefficients related?

    determinant is zero - no solutions, or โˆž solutions. determinant is not zero - 1 solution. If determinant is 1 - 1 solution ๐Ÿ’–

  6. [Matrix of coefficients of the linear system of equations] How would you go about proving that if determinant is zero, then system of linear equations either has: no solutions, or โˆž solutions.

    Consider a pair of linear equations Suppose that ad - bc = 0 (because determinant is 0!) Show that cx + dy are equal to 2 different values. Zero determinant (ad = bc) creates a dependence among coefficients that leads to bf = ed down the line.

  7. [Determining whether simultaneous equations have unique solutions] Is there a unique solution?