Matrix: Overview, Questions, Preparation

An invertible matrix is a square matrix that has an inverse, meaning there exists another matrix (its inverse) that, when multiplied by the original matrix, results in the identity matrix. In simpler terms, it is a matrix that can be "undone" by another matrix. 

Matrix operations involve performing mathematical calculations on matrices. Common operations include addition, subtraction, and multiplication. Addition and subtraction require matrices of the same dimensions, while multiplication has specific dimension requirements. 

We must remember these important points to avoid mistakes in our regular CBSE Exams.

• Always denote a matrix by capital letters e.g., $A,B,C,\dots \dots ,X,Y,Z$ and elements by small letters a,b,c,.......x,y,z.
• A matrix having m rows and n columns is called a matrix of order $m\times n$ or simple $m\times n$ (read as an m by n matrix).
• An element occurring in the i^th row and j^th column of a matrix A is called ${\left(i,j\right)}^{th}$  element of A and is denoted by $a_{ij}$ .
• The number of rows is written first and then the number of columns.

For example,

• • $\left[\begin{array}{cc}\hfill 3\hfill & \hfill 5\hfill \\ \hfill 7\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill -4\hfill \end{array}\right]$ is a matrix of order $2\times 3$             
  • $\left[\begin{array}{ccc}\hfill 7\hfill & \hfill 3\hfill & \hfill 0\hfill \\ \hfill 5\hfill & \hfill 4\hfill & \hfill 1\hfill \end{array}\right]$ is a matrix of order $3\times 2$
  • $\left[\begin{array}{cc}\hfill x^2\hfill & \hfill x\hfill \\ \hfill 5\hfill & \hfill 1\hfill \end{array}\right]$  is a matrix of order $2\times 2$ .
• In general, an $m\times n$ matrix has the following rectangular array:

or $A={\left[a_{ij}\right]}_{m\times n}$ or ${\left[a_{ij}\right]}_{m\times n}$  or $A={\left[a_{ij}\right]}_{m\times n}$ $1\le i\le m$   , $1\le j\le n$ $i,j\in N$ .

• The number of elements is an $m\times n$ matrix is equal to mn.