Matrix equation Ax=b

  1. Matrix Equation Ax=b Overview:
  2. Interpreting and Calculating AxAx
    • Product of AA and xx
    • Multiplying a matrix and a vector
    • Relation to Linear combination
  3. Matrix Equation in the form Ax=bAx=b
    • Matrix equation form
  4. Solving x
    • Matrix equation to an augmented matrix
    • Solving for the variables
  5. Properties of Ax
    • Addition and subtraction property
    • Scalar property
  1. Computing Ax
    Compute the following. If it cannot be computed, explain why:
    1. computing Ax
    2. compute Ax
    3. calculating Ax
  2. Converting to Matrix Equation and Vector Equation
    Write the given systems of equations as a vector equation, and then to a matrix equation.
    6x1+2x23x3=1 6x_1+2x_2-3x_3=1
    2x15x2+x3=4 2x_1-5x_2+x_3=4
    x12x27x3=5 -x_1-2x_2-7x_3=5
    1. Solving the Equation AX=bAX=b
      Write the augmented matrix for the linear system that corresponds to the matrix equation Ax=bAx=b. Then solve the system and write the solution as a vector.
      1. Solving the Equation Ax=b
      2. solve Ax=b
    2. Ax=b with unknown b terms
      Let finding b in Ax=b and finding b in Ax=b. Show that the matrix equation Ax=bAx=b does have solutions for some bb, and no solution for some other bb's.
      1. Understanding Properties of Ax
        Recall that the properties of the matrix-vector product Ax is:

        If AA is an m×nm \times n matrix, uu and vv are vectors in Rn\Bbb{R}^n, cc is a scalar, then:
        1. A(u+v)=Au+Av A(u+v)=Au+Av
        2. A(cu)=c(Au) A(cu)=c(Au)

        Using these properties, show that:
        A[(2u3v)(2u+3v)]=4A(u2)9A(v2) A[(2u-3v)(2u+3v)]=4A(u^2 )-9A(v^2)