CBSE Class 12 Commerce Mathematics Matrices
CBSE Class 12 Commerce Mathematics Matrices:-Introduction
CBSE Class 12 Commerce Mathematics MatricesCBSE Class 12 Commerce Mathematics Matrices:-The knowledge of matrices is necessary in various branches of mathematics. Matrices are one of the most powerful tools in mathematics. This mathematical tool simplifies our work to a great extent when compared with other straight forward methods. The evolution of concept of matrices is the result of an attempt to obtain compact and simple methods of solving system of linear equations. Matrices are not only used as a representation of the coefficients in system of linear equations, but utility of matrices far exceeds that use. Matrix notation and operations are used in electronic spreadsheet programs for personal computer, which in turn is used in different areas of business and science like budgeting, sales projection, cost estimation, analysing the results of an experiment etc. Also, many physical operations such as magnification, rotation and reflection through a plane can be represented mathematically by matrices. Matrices are also used in cryptography. This mathematical tool is not only used in certain branches of sciences, but also in genetics, economics, sociology, modern psychology and industrial management.
CBSE Class 12 Commerce Mathematics Matrices:-Matrix
Definition 1:- 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.
CBSE Class 12 Commerce Mathematics Matrices:-Order of a matrix
A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix). So referring to the above examples of matrices, we have A as 3 × 2 matrix, B as 3 × 3 matrix and C as 2 × 3 matrix.
CBSE Class 12 Commerce Mathematics Matrices:-Types of Matrices
- Column matrix
- A matrix is said to be a column matrix if it has only one column.
- Row matrix
- A matrix is said to be a row matrix if it has only one row.
- Square matrix
- A matrix in which the number of rows are equal to the number of columns, is said to be a square matrix.
- Diagonal matrix
- A square matrix B = [bij] m × m is said to be a diagonal matrix if all its non diagonal elements are zero, that is a matrix B = [bij] m × m is said to be a diagonal matrix if bij = 0, when i ≠ j.
- Scalar matrix
- A diagonal matrix is said to be a scalar matrix if its diagonal elements are equal, that is, a square matrix B = [bij] n × n is said to be a scalar matrix if bij = 0, when i ≠ j bij = k, when i = j, for some constant k.
- Identity matrix
- A square matrix in which elements in the diagonal are all 1 and rest are all zero is called an identity matrix.
- Zero matrix
- A matrix is said to be zero matrix or null matrix if all its elements are zero.
CBSE Class 12 Commerce Mathematics Matrices:-Equality of matrices
Definition 2 :-Two matrices A = [aij] and B = [bij] are said to be equal if
(i) they are of the same order
(ii) each element of A is equal to the corresponding element of B, that is aij = bij for all i and j.
CBSE Class 12 Commerce Mathematics Matrices:-Properties of matrix addition
The addition of matrices satisfy the following properties:
- Commutative Law If A = [aij], B = [bij] are matrices of the same order, say m × n, then A + B = B + A. Now A + B = [aij] + [bij] = [aij + bij]
= [bij + aij] (addition of numbers is commutative)
= ([bij] + [aij]) = B + A - Associative Law For any three matrices A = [aij], B = [bij], C = [cij] of the same order, say m × n, (A + B) + C = A + (B + C).
Now (A + B) + C = ([aij] + [bij]) + [cij]
= [aij + bij] + [cij] = [(aij + bij) + cij]
= [aij + (bij + cij)] (Why?)
= [aij] + [(bij + cij)] = [aij] + ([bij] + [cij]) = A + (B + C) - Existence of additive identity Let A = [aij] be an m × n matrix and O be an m × n zero matrix, then A + O = O + A = A. In other words, O is the additive identity for matrix addition.
- The existence of additive inverse Let A = [aij]m × n be any matrix, then we have another matrix as – A = [– aij]m × n such that A + (– A) = (– A) + A= O. So – A is the additive inverse of A or negative of A.
CBSE Class 12 Commerce Mathematics Matrices:-Properties of scalar multiplication of a matrix
If A = [aij] and B = [bij] be two matrices of the same order, say m × n, and k and l are scalars, then
- k(A +B) = k A + kB, (ii) (k + l)A = k A + l A
- k (A + B) = k ([aij] + [bij])
= k [aij + bij] = [k (aij + bij)] = [(k aij) + (k bij)]
= [k aij] + [k bij] = k [aij] + k [bij] = kA + kB - ( k + l) A = (k + l) [aij]
= [(k + l) aij] + [k aij] + [l aij] = k [aij] + l [aij] = k A + l A
CBSE Class 12 Commerce Mathematics Matrices:-Properties of multiplication of matrices
The multiplication of matrices possesses the following properties, which we state without proof.
- The associative law For any three matrices A, B and C. We have (AB) C = A (BC), whenever both sides of the equality are defined.
- The distributive law For three matrices A, B and C.
- A (B+C) = AB + AC
- (A+B) C = AC + BC, whenever both sides of equality are defined.
- The existence of multiplicative identity For every square matrix A, there exist an identity matrix of same order such that IA = AI = A.
CBSE Class 12 Commerce Mathematics Matrices:-Transpose of a Matrix
Definition 3:- If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A is called the transpose of A. Transpose of the matrix A is denoted by A′ or (AT). In other words, if A = [aij]m × n, then A′ = [aji]n × m.
CBSE Class 12 Commerce Mathematics Matrices:-Symmetric and Skew Symmetric Matrices
Definition 4:- A square matrix A = [aij] is said to be symmetric if A′ = A, that is, [aij] = [aji] for all possible values of i and j.
Definition 5:-A square matrix A = [aij] is said to be skew symmetric matrix if A′ = – A, that is aji = – aij for all possible values of i and j. Now, if we put i = j, we have aii = – aii. Therefore 2aii = 0 or aii = 0 for all i’s.
CBSE Class 12 Commerce Mathematics Matrices:-Elementary Operation (Transformation) of a Matrix
There are six operations (transformations) on a matrix, three of which are due to rows and three due to columns, which are known as elementary operations or transformations.
- The interchange of any two rows or two columns. Symbolically the interchange of ith and jth rows is denoted by Ri ↔ Rj and interchange of ith and jth column is denoted by Ci ↔ Cj.
- The multiplication of the elements of any row or column by a non zero number. Symbolically, the multiplication of each element of the ith row by k, where k ≠ 0 is denoted by Ri → k Ri.
The corresponding column operation is denoted by Ci → kCi - The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non zero number.
Symbolically, the addition to the elements of ith row, the corresponding elements of jth row multiplied by k is denoted by Ri → Ri + kRj.
Recommended Read:- CBSE Class 12 Commerce Mathematics Matrices
CBSE Class 12 Commerce Mathematics Matrices
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