In today’s blog, I will review some very basic results in 2×2 and 1×2 matrices.
1. Matrix defined
A matrix is a grouping of numbers that allows working on all the numbers at the same time.
For example, let’s consider a 2 x 2 matrix that can be based on a set of numbers: 1, 2, 3, 4.
2. Addition and subtraction of matrices
3. Multiplication of Numbers with Matrices
4. Product of Two Matrices
Now, here’s where it gets a bit confusing. We normally refer to a matrix using a capital letter. So let’s say we have two matrices A,B such that: A is a 2×2 matrix and B is a 2×2 matrix. We cannot assume that AB = BA. For example, if we reverse the matrices above, we get the following equation:
Another important point is that there is no product defined for a 2×1 matrix and a 2×2 matrix or a2x2 matrix and 1×2 matrix (since order is important in matrix products) and for that matter, there is no product defined a 2×2 matrix with a 1×2 matrix. In the case of 2×2 matrices, you can only get a product for a 2×2 matrix with a 2×2 matrix or a 1×2 matrix with a 2×2 matrix.
Lemma 1: det(AB) = (detA)(detB)
(3) det(AB) = (ae+bg)(cf+dh) – (af+bh)(ce+dg) = (acef + adeh + bcfg + bdgh) – (acef + adfg + bceh + bdgh) = adeh + bcfg – adfg – bceh.
(4) det(A) = ad – bc
(5) det(B) = eh – fg
(6) So det(A)det(B) = (ad – bc)(eh – fg) = adeh + bcfg – adfg – bceh
6. Identity Matrix
Lemma 2: AI = IA = A
We denote the inverse of A as A-1 and we define it as:
Lemma 2: AA-1 = A-1A = I
Lemma 3: det A-1 = 1/(det A)
(1) (det A)(det A-1) = det(AA-1) [From Lemma 1]
(2) det(AA-1) = det(I) [From Lemma 2]
(3) det(I) = 1*1 – 0*0 = 1. [Definition of I, Definition of Determinant]
(4) So, (det A)(det A-1) = 1
(5) And dividing both sides by (det A) gives us:
det A-1 = 1/(det A)
7. Final Points
The last point here is that while AA-1 = I, it is not necessarily true that ABA-1 = B. The reason is that AB does not necessarily equal BA and we are not allowed to change the order of the matrix elements.
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Thank you !