Topics are arranged as per SYLLABI published by Maharashtra State Board of Secondary and Higher Secondary Education.

A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix. A matrix having m rows and n columns is called matrix of order m x n.

Operations performed on rows or columns of a matrix are called Elementary Transformation on a Matrix. There are six operations (3 for rows and 3 for columns) which are known as elementary transformation:

- Interchanging any two rows (or columns)
- Multiplication of the element of any row (or column) by a non-zero scalar quantity.
- Addition of constant multiple of the elements of any row (or column) to the corresponding element of any other row (or column).

- The minor of any element of determinant of a matrix A, is the determinant obtained by deleting the row and the column in which that element is lying.
- The co-factor of an element is the minor of with proper sign.
- Co-factor is usually denoted by the corresponding capital letters, i.e. the co-factor of is denoted by

Elementary Transformations:

- Interchanging and row is denoted by .
- Multiplication of each element of row by non-zero
*k*is denoted by . - Addition to the elements of row, the corresponding elements of row multiplied by non-zero
*k*is denoted by .

Elementary Transformations:

- Interchanging and column is denoted by .
- Multiplication of each element of column by non-zero
*k*is denoted by . - Addition to the elements of column, multiplied by non-zero
*k*is denoted by .

- A rectangular matrix does not possess inverse matrix.
- Inverse of every square matrix, if exists, is unique.
- A square matrix is invertible if and only if it is non-singular.
- The inverse of any non-singular matrix can be obtained by elementary row transformations or by elementary column transformations.
- By doing one more elementary operations if we obtain all in one or more rows of left side matrix of , then does not exist.
- If A is a square matrix with n-rows; it is said to be invertible if there exists an n-rowed matrix B such that AB = BA = I
_{n}[where I_{n}is the unit matrix for order n]. B is called inverse of A or Reciprocal of A. - The inverse of the inverse is the original matrix itself .
- If A is a non-singular matrix, then
- The inverse of the transpose of a matrix is the transpose of its inverse.
- If A and B are two invertible matrices of the same order then AB is also invertible and .
- If A, B, C are square matrices of the same order 'n'; If A is a non-singular matrix, then
- Left Cancellation Law: AB = AC B = C
- Right Cancellation Law: BA = CA B = C

- If A is a non-singular matrix such that A is symmetric, then is also symmetric.

Let us assume; A be a square matrix of order 'n'. Then (or )

Let a sequence of elementary row (column) operation reduce A on the left (right) side to I_{n}. The same elementary row (column) operation reduces I_{n} on the left (right) side to a matrix B. Then I_{n} = BA.

Multiplying both sides by we get .

Using Associative Law we get .

We know that . So .

- The adjoint of a square matrix A is the transpose of the matrix of co-factors of matrix A.
- Adjoint of A is written as adj. A
- Adjoint of a diagonal matrix is a diagonal matrix.
- If A and B are two square matrices of the same order, then .
- ; where A is a non-singular matrix.
- ; where A is a non-singular matrix.
- The inverse of any non-singular square matrix can be obtained by using:

The matrix form of these equations is :

It is in the form of AX = B ; where matrix A is the prefactor of X.

To find X, we perform row operation on A to reduce A to I. We perform same row operation on B and let B get transformed to B_{1}.

Here AX = B gets transformed into IX = B_{1} form here we can obtain values of X.

Let’s go back to original condition, where AX=B.

Let’s assume A is the matrix of three equations with three unknowns.
Instead of reducing A to I by means of row operation, we sometimes reduce A to a matrix in which first column is having elements 1, 0, 0 in the order and there at least one of the c, d, e and f is zero.

In the equation (in above section) AX = B . Let ; so that exists uniquely. Pre-multiplying both sides by we get

On simplification we get .

- If , then system of equations is consistent and has a unique solution.
- If , and (adj A) B = 0 ; Then the system of equations is consistent and has many solutions.
- If , and (adj A) B 0 ; Then the system of equations is inconsistent (has no solutions).
- If all the constants, b
_{1}, b_{2}.... b_{n}; are all zero; then system of equation is said to be homogenous. Here AX = O; where O is a null matrix.- If ; then its only solution X=0 is called the trivial solution.
- If , then AX=O have a non-trivial solution. It will have infinite solutions.

- The square matrix A is defined to be singular if and only if .
- The square matrix A is defined to be non-singular if and only if .
- Determinant of matrix A is denoted by det. A or .
- is always a symmetric matrix.
- is always a skew – symmetric matrix.
- is always a symmetric matrix.
- Only a square matrix has a diagonal.
- If the determinants of two square matrices are equal, then those two matrices are need not be equal.
- If A is a square matrix, then the sum of all diagonal elements of A is called trace of A.
- A square matrix is said to be symmetric if element is the same as element. i.e. .
- A square matrix is said to be skew-symmetric if element is negative of its element. i.e. .
- Product AB is defined if and only if: Number of columns of A = Number of rows of B.
- Addition is defined only for matrices, which are of the same type.
- A non-zero matrix A is said to be in Echelon form , if A satisfies the following conditions:
- All the non-zero rows of A , if any precedes the zero rows.
- The number of zeros preceding the first non-zero element in a row is less than the number of such zeros in the succeeding row.
- The first non-zero element in a row is unity.

- Number of non-zero rows of a matrix given in the Echelon form is its rank.
- Addition is defined only for those matrices which are of the same type.
- Zero Matrix O is the additive identity, which is unique.
- A square matrix is an identity matrix if .
- Adjoint of a matrix of order 2 x 2 can be found by interchanging the elements of leading diagonal and changing the sign of the other two elements which are not in leading diagonal.
- Types of Matrices
- Row Matrix: Consists of only one row
- Column Matrix: Consists of only one column
- Square Matrix: Number of rows = Number of Columns Upper Triangular Matrix: Elements below principal diagonal are all zero
- Lower Triangular Matrix: Elements above principal diagonal are all zero.
- Diagonal Matrix: All non-diagonal elements are zero.
- Scalar Matrix: Diagonal Matrix with all elements of diagonal are equal.
- Unit Matrix (Identity Matrix) : Diagonal Matrix with all diagonal elements=1
- Zero Matrix (Null Matrix): Every element is zero
- Transpose of Matrix: Obtained by interchanging rows and columns.

- If co-factor of each element of a Matrix is kept at corresponding places then Matrix obtained is known as co-factor matrix.
- In real life matrices are used in Cryptography, Economics, Sociology, Modern Psychology and various applications in Industrial management.