Solving General Linear Equations w/ Excel

Matrix Operations in Excel Excel has commands for:

Multiplication (mmult)…matrix multiplication

Transpose (transpose)…transpose a matrix

Determinant (mdeterm)…calc the determinate

Inverse (minverse)…generate the matrix inverse

Important to remember that these commands apply to an array of cells instead of to a single cell

When entering the command, you must identify the entire array where the answer will be displayed!

Practice Problem Find AB = C

(2 X 4) x (4 X 3) = ?

Excel Matrix Multiplication

First enter the two matrices (for A and B):

Excel Matrix Multiplication Good practice to label the matrices:

Excel Matrix Multiplication Shading and borders help the matrices stand out:

A x B = C….what size matrix must C be?

Excel Matrix Multiplication An array of cells for the product must be selected –

in this case, a 2 × 3 array: Select the cells – directly type mmult(array1,array2). You couldn’t paste the equation into the selected cells, they are in different dimensions.

Excel Matrix Multiplication The MMULT function has two arguments: the

ranges of cells to be multiplied. Remember that the order of multiplication is important.

Excel Matrix Multiplication Using the Enter key with an array command only returns an

answer in a single cell. Instead, use Ctrl + Shift + Enter keys with array functions (This only works when you highlight the solution area, then put the cursor in the top ‘function typing area’)

Try this out!

Excel Matrix Multiplication Answer cells formatted:

Excel Transpose Use Ctrl + Shift + Enter to input command

Try this out! (This only works when you highlight

the solution area, then put the cursor

in the top ‘function typing area’)

Excel Determinant Since determinant is a scalar, select a single cell

and use Enter to input command

Excel Matrix Inversion Remember that only square matrices can have

inverses

Excel Matrix Inversion Ctrl + Shift + Enter to input command

A X A-1 = I, the identity matrix:

Try this out: make a matrix – get its inverse matrix –

multiply these two matrix to get the identity matrix

Solving Linear Equations - Excel Consider these 3 linear equations with unknowns x, y

and z:

Rewrite in matrix form.

MATLAB solution:

>> A=[12 32 -10; 0 2 3; 7 16 5];

>> b=[-30; 11; 42];

>> x=A\b

x =

7.0000

-2.0000

5.0000

Excel Solution Enter coefficient and constant matrices:

b =

Excel Solution Label and highlight cells for matrix of unknown

variables:

b =

Excel SolutionExcel does not have the convenient left division operator, so we must enter x = A-1b. Enter formula to invert Amatrix and multiply the result by the b matrix. This can be done in two steps or with nested commands:

b =

Excel Solution Apply formula to the selected array of cells by

pressing Ctrl + Shift + Enter:

b =

Exercises Solve the following sets of equations using

MATLAB and Excel and check the answer

0445

2

010423

zyx

zx

zyx

0425

02

23

02343

421

432

431

4321

xxx

xxx

xxx

xxxx

Solution 1

A = 3 -2 4 x B = 10

1 0 1 y 2

1 5 -4 z -4

x = 3.6

-2.8

-1.6

0445

2

010423

zyx

zx

zyx

Solution 2

0425

02

23

02343

421

432

431

4321

xxx

xxx

xxx

xxxx

A = 3 1 4 -1 x1 B = 23

3 0 1 1 x2 2

0 -2 1 1 x3 0

1 5 0 -2 x4 -4

x = 6.35

-8.52

-0.91

-16.13

So far we have been given different equations for a set of variables and solved for those variables:

2x1 + 3x2 = 12

4x1 – 3x2 = 6

Coefficients are known

x1, x2 are unknown

6

12

3- 4

3 2

2

1

x

x

It is common to have experimental measurements where the data are known but the equation or the equation coefficients are unknown.

Other uses of Ax=b and left-division

Fitting experimental data

Suppose someone is experimenting with a temperature controller. They apply a series of voltages and measure the resulting temperature.

Voltage Temperature

2 88.78

4 95.79

6 97.45

8 107.43

10 115.85

Fitting experimental data

You have reason to believe the relationship should

be linear, i.e.

T = c1V + c2

You know T, V

You want to find c1, c2

Notice this is the same equation as y = mx + b,

so we’re solving for slope and intercept.

Fitting experimental data First, let’s have a look at the data:

>> V = [2:2:10];

>> T = [88.78, 95.79, 97.45, 107.43, 115.85];

>> plot (V, T, ‘*’);

Not exactly a pretty straight line.

Could be

- The system really is not linear

- The voltage wasn’t exactly what

you thought

- The temperature measurement is

not accurate enough

- Etc.

Least Squares Fitting

• What line best represents the data?

x

y

• The best fit might not actually go through any points

• Forcing the line through a data point is risky

Least Squares Algorithm

• Least Squares minimizes the error between the line and the data points

• Find the line that minimizes the sum:

2)( i

ii mxy

ii mxy

x

y

Data

point

Line

Put our data into matrix form

c1 * 2 + c2 = 88.78

c1 * 4 + c2 = 95.79

c1 * 6 + c2 = 97.45

c1 * 8 + c2 = 107.43

c1 * 10 + c2 = 115.85

Voltage Temperature

2 88.78

4 95.79

6 97.45

8 107.43

10 115.85

We have one equation, c1V + c2 = T, and 5 measurements:

(m * x + b = y)(m * V + b = T)

Put our data into matrix form

2 14 16810

111

𝑐1𝑐2

=

88.7895.7997.45107.43115.85

c1 * 2 + c2 = 88.78

c1 * 4 + c2 = 95.79

c1 * 6 + c2 = 97.45

c1 * 8 + c2 = 107.43

c1 * 10 + c2 = 115.85

Now it is in a familiar

form, Ax=b

This is over-determined2 14 16810

111

𝑐1𝑐2

=

88.7895.7997.45107.43115.85

Before we said we wanted equal number of

equations and unknowns for a unique solution.

Here we have 5 equations and 2 unknowns.

This would be redundant if the points were all

exactly on a line. Since they are not, this finds the

“best” (least-squares) compromise.

This is over-determined2 14 16810

111

𝑐1𝑐2

=

88.7895.7997.45107.43115.85

We can use the same left-division operator to

solve for c1, c2. MATLAB recognizes the system

is over-determined and switches to a least-squares

fit algorithm.

x = A\b

A x b

Solve % least square fit

clear all

close all

V=[2:2:10];

T=[88.78,95.79,97.45,107.43,115.85];

plot(V,T,'*’);

A=[2 1;4 1;6 1;8 1;10 1];

TT=transpose(T);

xx=A\TT;

hold on

line_vs=linspace(0,10,10);

line_ts=line_vs*xx(1)+xx(2);

plot(line_vs,line_ts);

x =

3.2890

81.3260

Our x vector components are c1, c2

Least squares fit to V, T data

Other function types Other types of functions can be fit using the same

technique. Consider this data

t y

0.0 0.82

0.3 0.72

0.8 0.63

1.1 0.60

1.6 0.55

2.3 0.50

Fitting a non-linear functionA linear fit looks like this:

You decide a decaying exponential function might be better

Fitting a non-linear function

The equation we want to try is

y(t) = c1 + c2e-t

Fitting a non-linear function does not mean the

system of equations is non-linear.

We will solve for c1, c2. This equation is still linear

with respect to c1, c2.

Put into matrix form

1 𝑒−0

1 𝑒−0.3

1111

𝑒−0.8

𝑒−1.1

𝑒−1.6

𝑒−2.3

𝑐1𝑐2

=

0.820.720.630.600.550.50

c1 + c2e-0 = 0.82

c1 + c2e-0.3 = 0.72

c1 + c2e-0.8 = 0.63

…

Solve clear all

close all

A=[1 exp(0); 1 exp(-0.3); 1 exp(-0.8); 1 exp(-1.1);1 exp(-1.6); 1 exp(-2.3)];

b=[0.82;0.72;0.63;0.60;0.55;0.50];

x1=[0;0.3;0.8;1.1; 1.6; 2.3];

plot(x1,b,'*');% plot the scattering dots

hold on

x=A\b;

line_x=linspace(0,2.5,20);

line_y=x(1)+x(2)*exp(-line_x);

plot(line_x,line_y);

x =

0.4760

0.3413

1 𝑒−0

1 𝑒−0.3

1111

𝑒−0.8

𝑒−1.1

𝑒−1.6

𝑒−2.3

𝑐1𝑐2

=

0.820.720.630.600.550.50

A x by(t) = 0.476 + 0.3413 e-t

New fit

Summary Curve fitting is a commonly-needed operation and

can be accomplished in Matlab as an Ax=b problem.

MATLAB’s left-division operator can be used to find

an exact solution in the case of equal number of

equations and unknowns.

It can also be used to perform a least-squares fit in

the case of more equations than unknowns.

Be aware though that the underlying algorithm is not

the same.