Show that the matrices A = [1 2 -1 4] and B = [2/3 -1/3 1/6 1/6] are multiplicative inverses in M_2, 2.
1 answer:
Proof:
Let's Multiply them!
![AB = \left[\begin{array}{ccc}1&2\\-1&4\\\end{array}\right]\left[\begin{array}{ccc}2/3&-1/3\\1/6&1/6\\\end{array}\right]=\left[\begin{array}{ccc} 1(2/3) + 2(1/6) & 1(-1/3) + 2(1/6)\\-1(2/3) + 4(1/6)&-1(-1/3) + 4(1/6)\\\end{array}\right]](https://tex.z-dn.net/?f=AB%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%262%5C%5C-1%264%5C%5C%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%2F3%26-1%2F3%5C%5C1%2F6%261%2F6%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%201%282%2F3%29%20%2B%202%281%2F6%29%20%26%201%28-1%2F3%29%20%2B%202%281%2F6%29%5C%5C-1%282%2F3%29%20%2B%204%281%2F6%29%26-1%28-1%2F3%29%20%2B%204%281%2F6%29%5C%5C%5Cend%7Barray%7D%5Cright%5D)
If you don't remember how to multiply matrices, don't worry. In order to get AB we focus on the rows of A, and the columns of B.
- The first row of A is [1 2] and the first column of B is [2/3 1/6], if we do de dot multiplication we get 1(2/3) + 2(1/6) = 2/3 + 1/3 = 1.
- The first row of A is [1 2] and the second column of B is [-1/3 1/6]. Here we get 1(-1/3) + 2(1/6) = -1/3 + 1/3 = 0.
- The second row of A is [-1 4] and the first column of B is [2/3 1/6]. Here we get -1(2/3) + 4(1/6) = -2/3 + 2/3 = 0.
- The second row of A is [-1 4] and the second column of B is [-1/3 1/6]. Here we get -1(-1/3) + 4(1/6) = 1/3 + 2/3 = 1.
Then ![AB = \left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]](https://tex.z-dn.net/?f=AB%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%5C%5C%5Cend%7Barray%7D%5Cright%5D)
We also have to see what happens with
.
![BA = \left[\begin{array}{ccc}2/3&-1/3\\1/6&1/6\\\end{array}\right]\left[\begin{array}{ccc}1&2\\-1&4\\\end{array}\right] = \left[\begin{array}{ccc} (2/3)(1) + (-1/3)(-1) & (2/3)(2) + (-1/3)(4)\\(1/6)(1) + (1/6)(-1)&(1/6)(2) + (1/6)(4)\\\end{array}\right]= \left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]](https://tex.z-dn.net/?f=BA%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D2%2F3%26-1%2F3%5C%5C1%2F6%261%2F6%5C%5C%5Cend%7Barray%7D%5Cright%5D%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%262%5C%5C-1%264%5C%5C%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D%20%282%2F3%29%281%29%20%2B%20%28-1%2F3%29%28-1%29%20%26%20%282%2F3%29%282%29%20%2B%20%28-1%2F3%29%284%29%5C%5C%281%2F6%29%281%29%20%2B%20%281%2F6%29%28-1%29%26%281%2F6%29%282%29%20%2B%20%281%2F6%29%284%29%5C%5C%5Cend%7Barray%7D%5Cright%5D%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D1%260%5C%5C0%261%5C%5C%5Cend%7Barray%7D%5Cright%5D)
With this, we can conclude that

where
is the identity matrix. An by definition, that means that
and
are multiplicative inverses.
You might be interested in
Answer:
what grade yo.u in
Step-by-step explanation:
..
Answer:

Step-by-step explanation:
The question is simple. All we have to do is divide
by
. The answer to that is
.
I don’t know ????????????????
6.25
explanation: if you divide 6.25 by 5 you get 1.25, if you divide 17.40 by 12 it’s 1.45. Therefore you pay more.
Answer:
3/5 is 60/100 in it's simplest form.
Step-by-step explanation:
60 divided by 20 is 3 and 100 divided by 20 is 5.
So 3/5