There is no new mathematics material this week. You will be required to use Excel to do matrix calculations and solve linear systems of equations in Excel HW #2. We provide Excel Tutorials to help you.

An *array* is the same as a *matrix*: a rectangular block of numbers. Mathematicians call it matrix, computer scientists call it array.

An *array formula* in Excel is a formula that returns a matrix as a result.
The array formulas we use in this class are

**mmult**= matrix multiplication**minverse**= matrix inverse**+, -, ***= matrix addition, matrix subtraction, number * matrix**mtranspose**= matrix transpose (in section 6-3).

You enter array formulas the same way you enter other formulas, except for two things:

- You have to select the area where the result goes before you start typing.
- When you are done, you have to hit on a PC, or on a Mac (the command key on a Mac is that loop-de-loop thing next to the half-eaten apple).

If you forget either one of those things, there is nothing you can do except start over with entering the formula.

**Example:** Suppose you want to multiply a 2x2
matrix A, stored in A1:B2, with a 2x2 matrix B
stored in D1:E2. You want to store the result in G1:H2. You need to do the
following:

- Select the output area G1:H2
- Type in the formula:
**=mmult(A1:B2,D1:E2)** - Hit on a PC, or on a Mac

**Caution**: Excel lets you calculate **=A1:B2*D1:E2**, but that is **not** the
matrix product. That is the element-by-element product (C11 = A11*B11, instead
of C11 = A11*B11+A12*B21).

**Example:** Suppose you want to compute 2A+3B instead, and store
it in G1:H2 again. You do the following:

- Select G1:H2
- Type in the formula:
**=2*A1:B2+3*D1:E2** - Hit on a PC, or on a Mac

**Example:** Suppose you want to find the inverse of a 3x3
matrix A stored in A1:C3. (You can only find the inverse of a **square** matrix).
The inverse is again of size 3x3.

- Select a 3x3 space to hold the result, for example D1:F3
- Type in the formula:
**=minverse(A1:C3)** - Hit on a PC, or on a Mac

A system of linear equations is written in matrix notation as **Ax=b**. **A** is
the matrix of coefficients, **x** is the vector of unknowns, **b** is
the vector of right-hand sides. Formally, the solution is **x = A ^{-1}*b**.
That is the way Excel solves equations.

Suppose you want to solve the system

*3x + 4y = 1*

*x - 2y = 7*

You type in the matrix somewhere (for example in A2:B3), likewise for the right-hand side (for example in D2:D3). Then you need to

- Select a 2x1 space to hold the result, for example F2:F3
- Type in the formula:
**=mmult(minverse(A2:B3),D2:D3)** - Hit on a PC, or on a Mac.

The result (3,-2) shows up in F2:F3.

**Note:** a couple of the problems say "solve by Gauss-Jordan Elimination". Just ignore that, and solve the equations by whatever method you prefer.

- Go through the Excel Tutorial 2 spreadsheet or watch the video.
- Complete Excel HW 2
- If you did not take the Exam Policies Homework before Exam 1, you need to take it now and pass with a 100% score.
- Take Practice Exam 2
- Review the Exam Policies if necessary, and take Exam 2. The exam has a time limit of one hour.

Last Updated: Wednesday, August 5, 2015