# Category Archives: Matrices

## 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)

(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

QED

7. Inverse

We denote the inverse of A as A^{-1} and we define it as:

A^{-1} =

Lemma 2: AA^{-1} = A^{-1}A = I

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.

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Thank you !