Inverse of a Matrix
Please read our Introduction to Matrices first.
What is the Inverse of a Matrix?
The Inverse of a Matrix is the same idea as the reciprocal of a number:
Reciprocal of a Number
But we don't write 1/A (because we don't divide by a Matrix!), instead we write A-1 for the inverse:
(In fact 1/8 can also be written as 8-1)
And there are other similarities:
When you multiply a number by its reciprocal you get 1
8 × (1/8) = 1
When you multiply a Matrix by its Inverse you get the Identity Matrix (which is like "1" for Matrices):
A × A-1 = I
It also works when the inverse comes first: (1/8) × 8 = 1 and A-1 × A = I
Identity Matrix
Note: the "Identity Matrix" is the matrix equivalent of the number "1":
A 3x3 Identity Matrix
- It is "square" (has same number of rows as columns),
- It has 1s on the diagonal and 0s everywhere else.
- It's symbol is the capital letter I.
The Identity Matrix can be 2×2 in size, or 3×3, 4×4, etc ...
Definition
So we have a definition of a Matrix Inverse ...
The Inverse of A is A-1 only when:
A × A-1 = A-1 × A = I
Sometimes there is no Inverse at all.
2x2 Matrix
OK, how do we calculate the Inverse?
Well, for a 2x2 Matrix the Inverse is:
In other words: swap the positions of a and d, put negatives in front of b and c, and divideeverything by the determinant (ad-bc).
Let us try an example:
How do we know this is the right answer?
Remember it must be true that: A × A-1 = I
So, let us check to see what happens when we multiply the matrix by its inverse:
And, hey!, we end up with the Identity Matrix! So it must be right.
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