Introduction

• A matrix is a rectangular arrangement of numbers.

• The numbers in the matrix are called entities or elements.

• The horizontal rows of elements in the matrix are called rows.

• The vertical columns of elements in the matrix are called columns.

• If a matrix has m rows and n columns, its order is denoted as m x n.

  Example ➖

• Number of rows: 2
• Number of columns: 2
• Matrix order: 2 x 2
• Number of components: 4 (rows x columns)

Square Matrix

• A matrix is called a square matrix if it has the same number of rows and columns.

• A is a square matrix of order 3.
• 3 is the main diagonal
• 4 is the cross diagonal

Identity Matrix (Unit Matrix)

• An identity matrix is a square matrix where the main diagonal has only 1s.

Equality of Matrices

• Two matrices are considered equal if they have the same order and each element of one matrix is equal to the corresponding element of the other matrix.

Transpose of a Matrix

• If A = [aij] is an m × n matrix, the transpose of A is obtained by interchanging its rows and columns.

  Example:

Matrix Addition

• To add two matrices, they must have the same dimensions, which means that their order must be the same. Here is an example of how to add two matrices:
• To add matrices A and B of the same order, just add their corresponding entries. Let's consider the following matrices:

A =

[ 1 2 3 ]
[ 4 5 6 ]
         2x3

B =

[  7  8  9 ]
[ 10 11 12 ]
            2x3

The sum of these two matrices is:

A + B =

[  1+7  2+8  3+9 ]
[ 4+10 5+11 6+12 ]

which simplifies to:

[  8 10 12 ]
[ 14 16 18 ]
            2x3

This is how you can add two matrices.

Matrix Subtraction

• Matrix subtraction involves subtracting the corresponding elements of two matrices.
• It can only be performed on matrices of the same size or dimensions.
• The resulting matrix will have the same dimensions as the original matrices being subtracted.
• To subtract matrices A and B, we subtract their corresponding elements.
• Subtract the elements in the same positions to obtain the corresponding elements of the resulting matrix.
• Matrix subtraction is only possible when the matrices have the same dimensions.
• If the matrices have different dimensions, subtraction cannot be performed.

Matrix A:

[ 2  4 ]
[ 1  3 ]
        2x2

Matrix B:

[ 1  2 ]
[ 3  5 ]
        2x2

To subtract matrices A and B, we subtract their corresponding elements:

A - B = [ 2-1  4-2 ] = [ 1  2 ]
        [ 1-3  3-5 ]   [-2 -2 ]

So, the resulting matrix is:

[ 1  2 ]
[-2 -2 ]
        2x2

Matrix Multiplication

• Matrix multiplication is an operation performed on two matrices.
• It involves multiplying the corresponding elements of rows and columns to obtain the resulting matrix.
• The number of columns in the first matrix must be equal to the number of rows in the second matrix.
• The resulting matrix will have dimensions determined by the number of rows of the first matrix and the number of columns of the second matrix.
• The element at the intersection of row i and column j in the resulting matrix is calculated by multiplying corresponding elements of row i in the first matrix with column j in the second matrix and summing the products.
• Matrix multiplication is not commutative, meaning the order of multiplication matters.
• If matrix A has dimensions m x n and matrix B has dimensions n x p, the resulting matrix AB will have dimensions m x p.

Example 01 :

Example 02 :

Matrix A:

[ 1  2  1 ]
[ 2  3  2 ]
           2x3 

Matrix B:

[ 2  3 ]
[ 1  2 ]
[ 3  1 ]
        3x2

• A x B =

  (1x2) + (2x2) + (1x3) = 2+2+3 = 7

  (1x3) + (2x2) + (1x1) = 3+4+1 = 8

  (2x2) + (3x1) + (2x3) = 4+3+6 = 13

  (2x3) + (3x2) + (2x1) = 6+6+2 = 14

The resulting matrix AB is:

[  7  8 ]
[ 13 14 ]
         2x2

Properties of Matrix Multiplication

1. Matrix multiplication is not commutative: In general, multiplying matrix A by matrix B does not yield the same result as multiplying B by A (A·B ≠ B·A).
2. Matrix multiplication is associative: When multiplying three matrices A, B, and C, the result of multiplying A by the product of B and C is the same as multiplying the product of A and B by C (A·(B·C) = (A·B)·C).
3. Identity matrix: Multiplying any matrix A by the identity matrix I gives the same matrix A (I·A = A and A·I = A).
4. Distributive property: Matrix multiplication distributes over addition. Multiplying matrix A by the sum of matrices B and C is equivalent to separately multiplying A by B and A by C, and then adding the results together (A·(B + C) = A·B + A·C).

Determinant of a Matrix

• The determinant of a matrix can be calculated by considering the top row elements and their corresponding minors.
• Start by multiplying the first element of the top row by its minor, then subtract the product of the second element and its minor.
• Continue this pattern, alternating between addition and subtraction, for each element of the top row until all elements have been considered.
• This process simplifies the calculation of the determinant.

Inverse of a Matrix

• The inverse of a matrix is a matrix that, when multiplied with the original matrix, gives the identity matrix as the result.
• In simpler terms, if we have a matrix A and its inverse matrix A⁻¹, multiplying A with A⁻¹ gives the identity matrix.

Solving Simultaneous Linear Equations using Matrices

• Solving equations is a big part of math that deals with linear equations and their properties.
• Matrices are used to solve these types of equations.
• Matrices are rectangular arrays of numbers arranged in rows and columns.
• We can solve for the variables in the equations by doing different arithmetic operations with these matrices.
• This can be used with equations that have any number of variables, making it a useful tool in many areas of science and engineering.

ax + by = e
cx + dy = f

Above two equations can be converted to following matrix multiplication.

							[a b]          [x]      =  [e] 
							[c d]          [y]         [f]
									2x2                       2x2

Example 01 :