Find the matrix function X^-1(t) whose value at t is the...

Find the matrix function $X^{-1}(t)$ whose value at t is the inverse of the given matrix $X(t)$.\\ $X(t) = \begin{bmatrix} e^t & e^{8t} \\ e^t & 6e^{8t} \end{bmatrix}$\\ $X^{-1}(t) = \boxed{}$

Transcript

00:01 Hi there in this question we are given with the matrix matrix function x of t is equal to e power t e power 80 e power 80.

00:08 E power 80 now we have to find the inverse of this given matrix function that is x inverse of t so let's see how we'll do that.

00:18 So first suppose that x inverse of t is equal to a b c d then we know that x of t x of t x of t multiplied by x inverse of t will give us identity, that is one zero zero one for every values of t, for every values of t.

00:42 So here we can say that the given function x of t is actually given to be e power t, e power 80, e power t, nine e power 80.

00:54 And this is equal to this is multiplied by a, b c d which will give us one.

01:03 This is e power t.

01:03 This is equal.

01:03 Equal to we'll be having this is equal to one zero zero one.

01:09 So from here while multiplying this, we'll get the first equation to be a .e.

01:15 Power t plus c .e.

01:17 Power a .t is equal to 1.

01:19 Then the next equation that we'll be getting is that b .e.

01:23 Power t plus d .e.

01:25 Power 8t is equal to 0.

01:27 From here we can also say that b e power t is equal to negative d e power a t and then a e power t plus 9 c e power 8 t is equal to 0 and the next thing is b e power t plus 9 d e power 8 t is equal to 1 so we're getting in this manner so from equations from these equations that is from this equation and from this equation, we have that.

02:00 We can change this equation into b .e power t is equal to 1 .9 d .e.

02:10 Power 80.

02:11 So here we can say that while equating these two, let it be one, let it be two.

02:17 By equating those 1 and 2, we'll get that negative d .e.

02:20 Power 8t is equal to 1 minus 9 d .e.

02:23 Power 80 and from there and from there we'll get that this implies that 8d e power 80 is equal to 1 and from here we'll get that d is equal to 1 divided by 8 e power negative 80 okay so we got d as d as 1 divided by 8 e power negative 80.

02:51 Now similarly, we already have that b, we have that b, e power t equal to negative d, e power 8.

03:00 So b, we have b, e power t is equal to negative d, e power 8 t.

03:07 So from here we can say that b, e power t is equal to negative 1 divided by 8, e power negative 8, t multiplied by e power 8 to t.

03:19 So from here, this is equal to negative 1 divided by 8.

03:22 So this implies b is equal to negative 1 divided by 8 e power negative t.

03:29 So we got the value of b as well, b is equal to negative 1 divided by 8 e power negative t.

03:36 So these are the next two equations which we were having.

03:39 So for the first one, we can rewrite it as, we can rewrite it as or the second one...

Question

Find the matrix function $X^{-1}(t)$ whose value at t is the inverse of the given matrix $X(t)$.\\ $X(t) = \begin{bmatrix} e^t & e^{8t} \\ e^t & 6e^{8t} \end{bmatrix}$\\ $X^{-1}(t) = \boxed{}$

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