Thu Feb. 23

In class we covered the definition and method for deriving identity and inverse matrices. We learned to solve systems of equations like

2x + 3y = 8

4x - y = 10

Using matrices

begin{bmatrix} 2 & 3 \ 4 & -1 end{bmatrix} cdot begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} 8 \ 10 end{bmatrix}

We would then solve the system. Multiplying on the inverse on both sides

begin{bmatrix} 2 & 3 \ 4 & -1 end{bmatrix}^{-1} cdot begin{bmatrix} 2 & 3 \ 4 & -1 end{bmatrix} cdotbegin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} 2 & 3 \ 4 & -1 end{bmatrix}^{-1} cdot begin{bmatrix} 8 \ 10end{bmatrix}

We get

begin{bmatrix} 1 & 0 \ 0 & 1 end{bmatrix} cdotbegin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} frac{19}{7} \ frac{6}{7}end{bmatrix}

So, since begin{bmatrix} 1 & 0 \ 0 & 1 end{bmatrix} is the 2-by-2 identity matrix, we have

begin{bmatrix} x \ y end{bmatrix} = begin{bmatrix} frac{19}{7} \ frac{6}{7}end{bmatrix}

We also looked at solving systems of inequalities

Assignment:
6.4 on p. 331: 3, 6-10, 15, 17, 18
6.5 on p. 339: 5-9, 11-15