CBSE Class 12 Mathematics Matrices And Determinants Notes

 MATRIX : A matrix is a rectangular array of m*n numbers arranged in m rows and n columns.

* Row Matrix : A matrix which has one row is called row matrix. A=[a _{^{i j}}]1*n

* Column Matrix: A matrix which has one column is called column matrix. A=[a _{i j}]_m*1

* Square Matrix: A matrix in which number of rows are equal to number of columns, is called a square matrix A= [a _{i j}]_m*m

* Diagonal Matrix : A square matrix is called a Diagonal Matrix if all the elements, except the diagonal elements are zero.

* Scalar Matrix: A square matrix is called scalar matrix it all the elements, except diagonal elements are zero and diagonal element are equal.

* Identity or Unit Matrix: A square matrix in which all the non diagonal elements are zero and diagonal elements are unity is called identity or unit matrix.

* Null Matrix: A matrix in which all element are zero.

* Equal Matrices: Two matrices are said to be equal if they have same order and all their corresponding elements are equal.

* Transpose of matrix: If A is the given matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of a matrix.

* Properties of Transpose:
If A & B are matrices such that their sum & product are defined, then

* Symmetric Matrix: A square matrix is said to be symmetric if

A=A^T  i.e  if A=[a _{i j} ]_m*m , then a _{i j}=a _{j i for all i,j.} Also elements of the symmetric matrix are symmetric about the main diagonal.

* Skew symmetric Matrix : A square matrix is said to be skew symmetric if AT = - A.

Multiplication of a matrix by a scalar:

IfA = [aij] m × n is a matrix and k is a scalar, then kA is another matrix which is obtained by multiplying each element of A by the scalar k.

*Product of matrices: If A & B are two matrices, then product AB is defined, if no. of columns of A = no. of rows of B. Let A= [a _{i j}]_{m*n and B = [ b jk]n*p} then AB is the matrix C of the order mxp.To get (i.k)th element C_ik of the matrix C we take i th row of A and k th column of B, multiply them elementwise and take the sum of all these products

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