Table of Contents

## Matrix Introduction – Structure – Order of Matrix

## Types of Matrices

### Row, Column, Null, Square, Diagonal, Scalar, Identity, Triangular (Upper & Lower)

### Transpose of Matrix

### Symmetric Matrix & Skew Symmetric Matrix

### Singular Matrix & Equal Matrix with example

## Addition of Matrices

The addition of two or more matrices are possible if the order of the matrices are same. That means, A = [a_{ij}] and B = [b_{ij}] be two matrices of order m x n, then their addition is possible and the resultant matrix is of order m x n. Look at the below video.

## Scalar Multiplication of Matrix

If A = [a_{ij}] be m x n matrix and k be any real number then kA is again m x n matrix obtained by multiplying each element of A by k. The matrix kA is called the scalar multiple of A by k. Here k is generally called scalar. Look at the below video.

## Matrix Multiplication and its properties

Two matrices A and B are said to be conformable for multiplication if the number of columns of A is the same as the number of rows of B, that means A is matrix of order m x n and B is of order p x q then product of A & B is possible if and only if, n =q and resulting matrix will have order m x q. Symbolically, A_{m x n} B _{p x q} = C _{m x q}

## Inverse of Matrix

Inverse of Matrix for a matrix A is denoted by A^{-1}. The inverse of matrix is another matrix, which on multiplication with the given matrix gives the multiplicative identity.

For a matrix A, its inverse is A^{-1}, and A.A^{-1 }= A^{-1}·A = I, where I is the identity matrix. The matrix whose determinant is non-zero and for which the inverse matrix can be calculated is called an invertible matrix.

To find **A ^{-1} = adj(A)/|A|** where |A| ≠ 0 and A is square matrix

Before start calculating inverse matrix, we need to understand determinant, minors an co-factors of matrix.

### Determinant

A matrix’s determinant is the matrix’s single unique value representation. Any row or column of the provided matrix can be used to calculate the determinant of the matrix. The sum of the product of the elements and their co-factors in a given row or column of the matrix is the determinant of the matrix.

#### How to find determinant of given matrix

### Minor

The minor is defined for every element of a matrix. The minor of a particular element is the determinant obtained after eliminating the row and column containing this element.

#### How to calculate minors of 2 x 2 and 3 x 3 Matrix

### Co-factor

The co-factor of an element is calculated by multiplying the minor with -1 to the exponent of the sum of the row and column elements in order representation of that element.

**Co-factor of a _{ij} = (-1)^{i + j}× minor of a_{ij}**

#### How to find Co-factors of elements of 2 x 2 Matrix

#### How to find Co-factors of elements of 3 x 3 Matrix

### Steps to calculate Inverse of Matrix

Now, you can calculate the inverse of a matrix by following the given steps:

**Step 1:**Calculate the minors of all elements of A.**Step 2:**Then compute co-factors of all elements and write the co-factor matrix by replacing the elements of A by their corresponding co-factors.**Step 3:**Find the adjoint of A (written as adj A) by taking the transpose of co-factor matrix of A.**Step 4:**Multiply adj A by reciprocal of determinant.

For a matrix A, its inverse **A ^{-1} = |Adj A| / |A|**, where |A| must not zero