Matrices


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.

The matrix itself looks like this:

2. Addition and subtraction of matrices

Addition and subtraction of matrices are exactly the same as if you added and subtracted the numbers independently:

3. Multiplication of Numbers with Matrices

Multiplication with an integer just applies the integer to all the values involved so that:

4. Product of Two Matrices

In addition to these properites, matrices have there own special operations. The product of 2 matrices is a bit confusing. We define a product of a 1 x 2 matrix with a 2 x 2 matrix as the following:

We define a product a 2 x 2 matrix with a 2 x 2 matrix as the following:

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.

5. Determinant

A determinant is a value that is derived from a 2×2 matrix. Here is the definition:

Lemma 1: det(AB) = (detA)(detB)

(1) Let A =

Let B =

(2) AB =

(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

QED

6. Identity Matrix

The Identity Matrix is referred to as I and defined as:

Lemma 2: AI = IA = A

(1) Let A =

(2) AI =

(3) IA =

QED

7. Inverse

Let A =

We denote the inverse of A as A-1 and we define it as:
A-1 =

Lemma 2: AA-1 = A-1A = I

(1)

(2)


QED

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)

QED

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.

You are free to ask your doubts here :       Click here 
Thank you !

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About shivang1729

I am a young student who loves math . I like number theory and inequalities part the most , and preparing for Math Olympiads :)

Posted on Tuesday,April 3, 2012, in Matrices and tagged . Bookmark the permalink. Leave a comment.

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