Matrices

### Introduction to Matrices

A matrix (plural: matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. It's a powerful tool for organising and manipulating data. Each item in a matrix is called an element or an entry.

The order of a matrix specifies its size. It is given as (number of rows) × (number of columns). For example, a matrix with 2 rows and 3 columns is of order 2 × 3.

Example:

`A = [[1, 2], [3, 4]]`

This is a 2 × 2 matrix, also known as a square matrix because it has the same number of rows and columns.

### Matrix Addition and Subtraction

To add or subtract matrices, they must have the same order. The operation is performed by adding or subtracting the corresponding elements.

Process: Matrix Addition/Subtraction

If `A = [[a, b], [c, d]]` and `B = [[p, q], [r, s]]`, then:

A + B = [[a+p, b+q], [c+r, d+s]]

A - B = [[a-p, b-q], [c-r, d-s]]

Example:

If `P = [[5, 8], [2, 1]]` and `Q = [[3, 1], [4, 6]]`, then

`P + Q = [[5+3, 8+1], [2+4, 1+6]] = [[8, 9], [6, 7]]`

### Scalar Multiplication

Scalar multiplication involves multiplying a matrix by a single number (a scalar). Every element inside the matrix is multiplied by this scalar.

If `k` is a scalar and `A = [[a, b], [c, d]]`, then:

k A = [[ka, kb], [kc, kd]]

Example:

If `A = [[3, -1], [0, 4]]`, then `5A = [[5*3, 5*(-1)], [5*0, 5*4]] = [[15, -5], [0, 20]]`

### Matrix Multiplication

Multiplying two matrices is more complex. To multiply matrix A by matrix B (to get AB), the number of columns in A must equal the number of rows in B.

Process: Matrix Multiplication (Row-by-Column)

If `A = [[a, b], [c, d]]` and `B = [[p, q], [r, s]]`, then:

AB = [[(ap+br), (aq+bs)], [(cp+dr), (cq+ds)]]

An important property is that matrix multiplication is not commutative, meaning that in general, AB ≠ BA.

### The Determinant of a 2x2 Matrix

The determinant is a special number that can be calculated from a square matrix. For a 2 × 2 matrix, it tells us important information, especially when finding the inverse.

The determinant of a matrix A is denoted as det(A) or |A|.

Formula: Determinant of a 2x2 Matrix

For a matrix `A = [[a, b], [c, d]]`:

det(A) = ad - bc

Example:

If `M = [[4, 2], [1, 5]]`, then `det(M) = (4)(5) - (2)(1) = 20 - 2 = 18`.

If the determinant of a matrix is zero, it is called a singular matrix. Singular matrices do not have an inverse.

### The Inverse of a 2x2 Matrix

The inverse of a square matrix A, denoted as A⁻¹, is the matrix such that when you multiply it by A, you get the identity matrix, `I`. The identity matrix for a 2x2 is `[[1, 0], [0, 1]]`.

So, A A⁻¹ = A⁻¹ A = I.

A matrix only has an inverse if its determinant is not zero.

Process: Finding the Inverse of a 2x2 Matrix

Formula: Inverse of a 2x2 Matrix

For a matrix `A = [[a, b], [c, d]]`:

A⁻¹ = (1 / (ad-bc)) * [[d, -b], [-c, a]]

Example:

Find the inverse of `M = [[4, 2], [1, 5]]`.