Friday, July 25, 2014

THIS ARTICLE GIVES YOU ABOUT MATRICES AND DETERMINANTS – By MaddaliSwetha



Note: Basically matrix is not asked in qualitative aptitude test in any interviews  but these matrix subject is very tricky questions can be asked to shortlist for next round in interviews. Matrices are essential for solving large sets of simultaneous equations using a computer.

In mathematics, a matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The individual items in a matrix are called its elements or entries.  

1. Matrix: A set of numbers of real or complex arranged in the form of rectangular array having m rows and n columns is called m x n matrix. In other words, the number of rows and the number of columns is called order of a matrix there are m rows n columns. Hence, the order can be written as m x n or read as m by n.

2. Rectangular Matrix: If the number of rows and columns of a matrix are unequal the matrix is called rectangular matrix.

Example of a row matrix: [1, 2, 3] is written as 1 x 3 and it contains 1 row and 3 columns.

3. Column Matrix: A matrix having only one column matrix. Example: 3 rows and 1 column it is written as 3 x 1.

4. Null Matrix or Zero Matrix: If every element of a matrix is O then it is called a null matrix or a zero matrix. It is denoted by O m x n. Example: O 3 x 3 [It contains O rows and O columns.

5. Square Matrix: If number of rows is equal to number of columns is called square matrix.

6. Principal Diagonal: The diagonal of a square matrix from the first element. of first row to the last element of last row is called principal diagonal. The elements along the principal diagonal are called diagonal elements. Sum of the Diagonal elements is called Trace of a matrix.

7. Triangular Matrix: A Triangular matrix is a special kind of square matrix. 
A square matrix is called lower triangular if all the entries above the main diagonal are zero. 



A square matrix is called upper triangular if all the entries below the main diagonal are zero. 



i.e, A triangular matrix is one that is either lower triangular or upper triangular. 

A matrix that is both upper and lower triangular is called a diagonal matrix.



8. Identity Matrix or Unit Matrix:
The identity matrix in of size n is the n-by-n matrix in which all the elements on the main diagonal are equal to 1 and all other elements are equal to 0, e.g.


 9. Scalar Matrix: scalar matrix in which the diagonal elements are equal all other elements being zero's is called a scalar matrix.

10. Skew Symmetric matrix: changing rows into columns or columns into rows matrix remaining same then that matrix is skew symmetric matrix.

11. Additional Matrices: If A and B are two matrices of same type their sum is written as A+B.

Rules of Additional Matrices:
A+B = B+A
A+O = A
A+(-A) = O

12. Multiplication of Matrices: Two matrices are said to be conformable for multiplication if the number of columns in the first matrix are equal to the number of rows in the 2nd matrix.
Each matrix is of order 2 x 2 i.e., no of rows in one matrix is same as number of columns in the other matrix.

Rules of Multiplication of Matrices:

1. AB exists, BA need not necessarily exist.
2. AB = O it need not necessary imply that A = O, or B = O
3. A, B ,C are 3 matrices the A (B+C) = AB+BC addition and multiplication is assured. 
4. A(B-C) = AB - AC subtraction and  multiplication of matrices assured.

13. Notation:
Matrices are commonly written in box brackets:



An alternative notation uses large parentheses instead of box brackets:



The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are usually symbolized using upper-case letters (such as A in the examples above), while the corresponding lower-case letters, with two subscript indices (e.g., a11, or a1,1), represent the entries.

14. Difference between Matrix and Determinant?

Matrix: 
1. A Matrix cannot be reduced to a single number.
2. Number of rows and columns may be equal.
3. Interchange of rows and columns gives a different matrix.

Determinant:
1. A determinant can be reduced to a single number.
2. The number of rows and columns must be equal.
3. Interchange of rows and columns gives the same determinant with change of sign.

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