Enter 2x2 Matrix:
Finding the Inverse of a 2x2 Matrix: An Example
Imagine you have a secret code represented by a 2x2 matrix:
| 3 4 |
| 1 2 |
To decipher this code, you will need its secret key, which is also a 2x2 matrix, called the inverse. But how do you find it? Let's embark on a mathematical adventure to uncover the mystery!
Step 1: Formula
For a 2x2 matrix like ours, finding the inverse involves a special formula:
A^-1 = 1 / (ad - bc) * [ d -c ]
[ -B. A ]
Where:
* A^-1 is the inverse matrix
* A is the original matrix (our secret code)
* A, B, C and D are elements of A.
Step 2: Decoding the secret code
In our case, we have:
A = 3, B = 4, C = 1, and D = 2
Plugging these values into the formula:
A^-1 = 1 / (3*2 - 4*1) * [2 -1 ]
[ -4 3 ]
A^-1 = 1/2 * [2 -1 ]
[ -4 3 ]
Step 3: Simplifying the Key
To make it easier to use our key, let's divide both sides by 2:
A^-1 = [1 -0.5 ]
[-2 1.5 ]
Step 4: Applying the Key and Revealing the Secret
Now, multiply the original matrix (the encoded message) by its inverse (the key):
A * A^-1 = [3 4 ] * [1 -0.5 ] = [1 0 ]
[1 2] [-2 1.5] [0 1]
Voila! The encrypted message, represented by the matrix product, reveals its true nature:
| 1 0 |
| 0 1 |
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