Inverse of 2 x 2 Matrix
Inverses of Matrices - How it Works - Video
Inverse Indenties
Inverse Identities:
For matrix to have an identity, the matrix must be a square. For example, it must be 2 x 2, 3 x 3, 4 x 4, 5 x 5, and so on.
Each identity is the diagonal from top left to bottom right contains ones and the rest are zeros.
Like in multiplication where if we multiply by the multiplicative identity the answer will be 1. For example 5 * (1/5) is 1. When we multiply, a matrix and it's inverse matrix the resultant matrices will just have the number 1 inside.
Example 1 - Step 1
Example 1 - Step 1:
Here we have added the identity matrix to make an augmented matrix.
We replaced R2 with 4 * R1 - * R2 and the reason is that we want the element a2,1 or 4 in this case to be 0.
So we multiplied the first row by 4 to that so our new R1 is 4, 8, 4, 0. Then we subtract our new R1 and R2 to get 0, 2, 4, -1.
We need our principle diagonal (top left to bottom right) to be only 1s and our secondary diagonal (bottom left to top right) to be 0s.
Example 1 - Step 2
Example 1 - Step 2:
We replaced R1 with R1 - * R2 and the reason is that we want the element a1,2 or 2 in this case to be 0.
So we subtract our R1 and R2 to get 1, 0, -3, 1.
Example 1 - Step 3
Example 1 - Step 3:
We replaced R2 with (1/2) * R2 and the reason is that we want the element a2,2 or 2 in this case to be 1.
So we multiply each element in R2 by (1/2) to get 0, 1, 2, -1/2.
Example 1 - Answer
Example 1 - Answer:
Now the numbers are after vertical line is the inverse of original matrix. So in R1 we have -3 and 1, and in R2 we have 2 and (-1/2).