Matrices Class 12 Notes Mathematics Chapter 3

CHAPTER AT A GLANCE

• Matrices A rectangular arrangement of numbers in rows and columns, is called a Matrix. This arrangement is enclosed by small ( ) or big [ ] bracket. A matrix is represented by capital letters A, B, C etc. and its element are by small letters a, b, c, x, y, etc.
• Order of a matrix A matrix which has m rows and n columns is called a matrix of order m × n.3.
• Types of matrices:

(i) Row Matrix – If in a matrix, there is only one row, then it is called a RowMatrix.

(ii) Column Matrix – If in a Matrix, there is only one column, then it is calleda column matrix.

(iii) Square Matrix – If number of rows and number of column in a matrix are equal, then it is called a square matrix.

(iv) Singleton Matrix – If in a matrix there is only one element, then it is called singleton matrix.

(v) Null or Zero Matrix – If in a matrix all the elements are zero, then it is called a null or zero matrix and it is generally denoted by O.

(vi) Diagonal Matrix – If all elements except the principal diagonal in a squarematrix are zero, it is called a diagonal matrix. Thus a square matrix, A = [aij] is a diagonal matrix if aij = 0, when i ¹ j

(vii) Scalar Matrix – If all the elements of the diagonal of a diagonal matrix are equal, it is called a scalar matrix.

(viii) Unit Matrix – If all elements of principal diagonal in a diagonal matrix are 1, then it is called unit matrix. A unit matrix of order n × n is denoted byIn.

(ix) Equal Matrices – Two matrices A and B are said to be equal matrix, if they are of same order and their corresponding elements are same.

• Addition and subtraction of matrices :

If A = [aij]m×n and B = [bij]m×n are two matrices of the same order then their sum A + B is a matrix whose each element is the sum of corresponding elements of two matrices.

Matrix addition and subtraction can be possible only when matrices are of same order.

• Properties of Addition of Matrices – If A, B and C are matrices of same order, then

(i) Commutative law : A + B = B + A

(ii) Associative law : (A + B) + C = A + (B + C)

(iii) A + O = O + A = A, where O is zero matrix of the order of matrix A. O is the additive identity of any matrix.

(iv) A + (–A) = O = (–A) + A , where ( –A) is obtained by changing the sign of every element of A. –A is the additive inverse of the matrix A

• Scalar multiplication of Matrix

Let A = [aij]m×n be a matrix then the matrix which is obtained by multiplying every element of A by any scalar number k is called scalar multiple of A and it is denoted by

• Properties of Scalar Multiplication:

If A, B are matrices of the same order and p, q are any two scalars then –

• Multiplication of Matrices

If A and B be any two matrices, then their product AB will be defined only when number of column in A is equal to the number of rows in B. If A = [aij]m×n and B = [bij]n×p, then their product AB = C = [cij], will be a matrix of order m × p, where,

• Properties of Matrix Multiplication – If A, B and C are three matrices such that their sum and product are defined, then
• Positive Integral Powers of a Matrix

The positive integral powers of a matrix A are defined only when A is a square matrix. Also

• Tanspose of a Matrix

The matrix obtained from a given matrix A by changing its rows into columns or columns into rows is called transpose of matrix A and is denoted by AT or A’. If order of A is m × n, then order of AT is n × m

• Properties of Transpose of a matrix –
• Symmetric and Skew-symmetric Matrix
• Properties of Symmetric and Skew-Symmetric Matrices –
• Elementary Row (Column) Operations
• Inverse of a matrix by elementary operations

Let X, A and Be be matrices of, the same order such that X = AB. In order to apply a sequence of elementary row operations on the matrix equation X = AB, we will apply these row operations simultaneously on X and on the first matrix A of the product AB on RHS.Similarly, in order to apply a sequence of elementary column operations on the matrix equation X = AB, we will apply, these operations simultaneously on X and on the second matrix B of the product AB on RHS.In view of the above discussion, we conclude that if A is a matrix such that A–1 exists, then to find A–1 using elementary row operations, write A = IA and apply a sequence of row operation on A = IA till we get, I = BA. The matrix B will be the inverse of A. Similarly, if we wish to find A–1 using column operations, then, write A = AI and apply a sequence of column operations on A = AI till we get, I = AB. The matrix B will be the inverse of A.

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