Matrices

1. Types of Matrices & Core Definitions

Singular Matrix

A matrix whose determinant is zero.

Inverse does not exist.

Non-Singular Matrix

A matrix whose determinant is non-zero.

Inverse exists.

Transpose of a Matrix �

Rows become columns and columns become rows.

Example:

Reversal Law of Transpose

Symmetric Matrix

Elements on both sides of the main diagonal are mirror images.

Example:

Skew-Symmetric Matrix

Example:

MHT CET Trick

In every skew-symmetric matrix:

All main diagonal elements are always zero.

2. Adjoint of a Matrix �

The adjoint of a matrix is the transpose of its cofactor matrix.

Property 1

Property 2 (Very Important)

where � = order of the matrix.

For a � matrix:

Property 3

Property 4

3. Inverse of a Matrix �

The inverse is a matrix that satisfies:

Main Formula

Condition: �

Property 1

Property 2 (Reversal Law)

Property 3

Property 4

Transpose and inverse can be interchanged.