I am trying to use the variable t instead of x in the matrix $M below and can't seem to figure out how to do so and I am not sure why 'x' works. Any help would be great.

## DBchapter(Operation calculus)

## Level(3)DOCUMENT();

loadMacros(

"PGstandard.pl",

"PGML.pl",

"MathObjects.pl",

"MatrixUnits.pl",

"VectorListCheckers.pl",

"PGchoicemacros.pl",

"contextFraction.pl",

"parserRadioButtons.pl",

#"PGchoicemacros.pl",

"unionLists.pl",

"PGcourse.pl",

);

Context("Matrix");

$S = Matrix([[Compute(random(4,6)),0,0],

[0,Compute(non_zero_random(2,6)),0],

[0,0,Compute(non_zero_random(-10,2))]]);

$s11=$S->element(1,1);

$s22=$S->element(2,2);

$s33=$S->element(3,3);

$M = Matrix(["e^(x*$s11)",0,0],

[0,"e^(x*$s22)",0],

[0,0,"e^(x*$s33)"]);

BEGIN_PGML

Recall that

[`` e^{t} = 1 + t + \frac{t^2}{2!} + \frac{t^3}{3!}+\cdots ``]

With this it seems reasonable to define

[`` e^{Ax} = I + xA + \frac{(xA)^2}{2!} + \frac{(xA)^3}{3!}+\cdots ``]

where [` x `] is a scalar.

Suppose that [` A `] is diagonlizable with eigenvalues [` \lambda_1=[$s11], \lambda_2=[$s22], \lambda_3=[$s33]`] so that

[` A `] is given by

[``A = W [$S] W^{-1}``]

where the columns of [` W `] are the eigenvectors of [` A `]: [` v_1, v_2, v_3 `].

Check to see that

[` (x A)^n = W \left[ \begin{array}{ccc}

([$s11] x)^n & 0 & 0 \\

0 & ([$s22] x)^n &0 \\

0 & 0 & ([$s33] x)^n

\end{array}

\right] W^{-1} `]

Given that, what is:

[``e^{xA} = W``][____]*{$M}[``W^{-1}?``]

END_PGML

ENDDOCUMENT();