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Matrices and Determinants Matrices 1 MATRICES AND DETERMINANTS MATRICES 1. Matrix :- Arrangement of the elements in rows and column 2. Order of the matrix :- (number of rows) (number of columns) 3. Types Of Matrix 1. Zero matrix :- A matrix in which each element is zero, is called zero matrix or null matrix 2. Row matrix :- A matrix having only one row 3. Column matrix :- A matrix having only one column 4. Square matrix:- A matrix in which the number of rows is equal to the number of columns Principle diagonal: - The diagonal from top left to bottom right 5. Diagonal matrix :- A matrix in which all the elements except the principle diagonal elements are zero 6. Scalar matrix :- A diagonal matrix in which all the principle diagonal elements are equal 7. Unit matrix or Identity matrix :- A diagonal matrix in which each principle diagonal entries is one 8. Idempotent matrix :- A matrix is said to be idempotent if 9. Nilpotent: A square matrix is called nilpotent matrix if there exist a positive integer ‘n’ such that . If ‘m’ is the least positive integer such that , them ‘m’ is called the index of the nilpotent matrix. 10. Involutory matrix :- A matrix is said to be involutory if 11. Upper triangular matrix :- A matrix in which all the elements below the principle diagonal are zero 12. Lower triangular matrix :- A matrix in which all the elements above the principle diagonal are zero 4. Trace of a matrix: - Trace of a matrix is defined and denoted as ( ) sum of principal diagonal element. 5. Two matrices are said to be equal if they have the same order and the corresponding elements are also equal. 6. If ( ) ( ) be two matrices of order then 1. ( ) [scalar (k) multiplication of matrix] 2. ( ) [matrix addition] 3. ( ) [matrix subtraction] 7. Matrix multiplication :- If ( )be the matrix of order and ( )be the matrix of order then ( ) be the matrix of order , where is the sum of the product of the corresponding elements of row of A and column of B. 8. If are matrices then 1) ) need not be equal to ) 9. Transpose of a matrix :- A matrix obtained from interchanging the rows into columns. If is the matrix then transpose of A is denoted by or i.e., ( ) then ( ) 10. Orthogonal matrix :- A matrix is said to be orthogonal if 11. Symmetric matrix :- A matrix is said to be symmetric if 12. Skew-symmetric matrix :- A matrix is said to be skew-symmetric if (diagonal elements of skew-symmetric matrices are zero) KCET SPSM PU SCIENCE COLLEGE 1 MATRICES AND DETERMINANTS 13. If is a square matrix , then 1) is a symmetric matrix 2) is a Skew-symmetric matrix ) is a symmetric matrix 14. If are square matrix then is symmetric iff 15. Properties of matrices :- If and are three matrices then 1. ( ) 2. ( ) 3. ( ) 4. ( ) 5. ( ) Where 6. ( ) ( ) i.e., matrix multiplication is associative 7. is need not be true. i.e., matrix multiplication is not commutative DETERMINANTS 16. If ( ) is a square matrix of order 2 then (determinate of ) | | | | 17. If ( ) then | | ( ) ( ) ( ) 18. Properties Of Determinants 1. The value of a determinant is unaltered if its rows and columns are interchanged ( | | | |) 2. If two rows (or columns) of a determinant are interchanged then sign of the determinant changes 3. If in a determinant two rows (or columns) are identical then the value of the determinant is zero 4. If the element of any row (or column) are multiplied by k then the value of the determinant is multiplied by k Note: - | | | | where is a constant and is the order of the matrix 5. If in a determinant one row (or column) is multiple of another then the value of the determinant is zero 6. If each element in any row (or column) of a determinant is a sum of two terms then the determinant can be expressed as a sum of 2 determinants 7. If to the elements of any row (or column) of a determinant the same multiples of the corresponding elements of the other rows (or columns) of the determinant are added then the value of the determinant is unaltered 8. If and are the square matrices of same order then | | | || | 9. The value of a determinant of upper triangular matrix or lower triangular matrix is equal to product of the principal diagonal element. 10. Determinant of a skew symmetric matrix of odd order is zero and of even order is a non-zero perfect square 19. The minor of an element is the value of the determinant obtained by deleting row and column of the matrix. The minor of the element is denoted by 20. The cofactor of an element of a matrix is denoted and defined by ( ) 21. The cofactor matrix of a square matrix is the matrix obtained by replacing the elements of by its corresponding cofactors 22. The adjoint of the matrix is the transpose of the cofactor matrix and it is denoted by 23. If is any square matrix then ( ) ( ) | | KCET SPSM PU SCIENCE COLLEGE 2 MATRICES AND DETERMINANTS 24. If A is any square matrix and | | then | | | | where n is the order of the matrix 25. If | | then A is called singular matrix and If | | then A is called non-singular matrix 26. If and are square matrix and is non-singular then 27. If is non-singular then (inverse of ) ( ) | | 28. If ( ) then ( ) 29. If is non-singular then ( ) ( ) 30. If is non-singular then ( ) ( ) ( ) 31. If is non-singular then | | | | 32. Solving the system of linear equations , and 1) Cramers Rule :- If | | , | | , | | and | | then unique solution : If and solution is , and 2) Matrix method :- If [ ] [ ] and [ ] then unique solution : If and solution is 33. Note: 1) If | | then there exist unique solution and the system of equation is consistent. 2) If | | and ( ) , then system of equation has infinitely many solutions and the system of equation is consistent. 3) If | | and ( ) , then system of equation has no solutions and the system of equation is inconsistent. 34. If is a square matrix and is the identity matrix of same order then is called characteristic matrix 35. | | is called characteristic equation of A and its roots are called characteristic roots or Eigen values 36. The Eigen values of upper or lower triangular matrix are the principal diagonal elements. 37. If is a square matrix of order then characteristic equation is given by ( ) | | , Where ( ) is a trace of the matrix A. Note:- 1. If A is a square matrix of order n, then A will have at most n Eigen values 2. If A is a square matrix then A and A’ have the same Eigen values 3. A is singular if and only if 0 is an Eigen value of A. 4. Sum of the Eigen values is always equal to the trace of the matrix. 5. If are the Eigen values of then are the Eigen values of 38. Cayley-Hamilton theorem :- Every square matrices satisfies its characteristic equations KCET SPSM PU SCIENCE COLLEGE 3 MATRICES AND DETERMINANTS PROBLEMS 1) If A is a square matrix then A AT is 1) Symmetric 2) Skew Symmetric 3) A Scalar Matrix 4) A unit Matrix 2) If are matrices of same order then ( ) is 1) skew symmetric matrix 2) null matrix 3) symmetric matrix 4) unit matrix 3) The matrix A is both symmetric and skew symmetric, then 1) A is diagonal matrix 2) A is zero matrix 3) A is a square matrix 4) none of these 4) Matrices A and B will be inverse of each other only if 1) 2) 3) 4) 5) If P, Y and W are the matrices of order respectively, such that is defined, then 1) 2) 3) 4) 6) The total number of possible matrices of order 3 X 3 with each entry 2 or 0 is 1) 2) 3) 4) 7) If the matrix [ ] is skew symmetric matrix, then 1) 4 2) 0 3) 4) 10 8) The symmetric part of the matrix ( ) 1) ( ) 2) ( ) 3) ( ) 4) ( ) 9) If ( ) then if the value of is 1) 2) 3) 4) 10) If [ ] [ ] [ ] then the values of are respectively. 1) 1, 0, 1 2) 1, 1, 0 3) 0, 1, 1 4) 1, 1, 1 11) If ( ) ( ) then the value of 1) 2) 3) 4) 12) If A is an orthogonal matrix then 1) | | 2) | | 3) | | 4) none of these 13) If A = diag(a,b,c) then An is 1) abc 2) diag (na, nb, nc) 3) diag (anbncn) 4) anbncn 14) If ( ) then 1) ( ) 2) ( ) 3) ( ) 4) ( ) 15) If A is a square matrix such that then ( ) 1) 2) 3) 4) 16) If A is a square matrix such that then, ( ) ( ) 1) 2) 3) 4) KCET SPSM PU SCIENCE COLLEGE 4 MATRICES AND DETERMINANTS 17) On using elementary column operations in the following matrix equation ( ) ( ) ( ) we have, 1) ( ) ( ) ( ) 2) ( ) ( ) ( ) 3) ( ) ( ) ( ) 4) ( ) ( ) ( ) 18) If A,B are square matrices of order 3 X 3 such that |A| = -1, |B| = 3, then | | 1) 2) 3) 4) 19) If is singular then is 1) Singular 2) non-singular 3) symmetric 4) skew-symmetric 20) If and then 1) 2) 3) 4) 21) If are two matrices such that then 1) 2 2) 3) 4) 22) If [ ] then ( ) 1) ( ) 2) ( ) 3) ( ) 4) ( ) 23) If is a matrix of order such that ( ) then | | 1) 30 2) 40 3) 100 4) 10 24) If is the root of the equation then the value of | | is 1) 0 2) 1 3) 4) 25) If , , are the roots of = 0, then the value of the determinant | | is 1) 2) 3) 4) 26) The value of the determinant of a skew symmetric matrix of odd order is 1) Zero 2) Perfect square 3) Can’t be predicted 4) none of these 27) There are two values of a which makes determinant | | then sum of these numbers is 1)4 2) 5 3) -4 4) 9 28) If | | | | 1) 2) 2k 3) 3k 4) 6k 29) If | | then ( ) 1) ( ) 2) ( ) 3) ( ) 4) ( ) KCET SPSM PU SCIENCE COLLEGE 5 MATRICES AND DETERMINANTS 30) The value of | | = 1) abc 2) a + b + c 3) 0 4) 1 31) In triangle |( ) ( ) ( ) | , then the triangle is ( ) ( ) ( ) 1) Equilateral 2) Isosceles 3) Right angled 4) Scalene 32) The value of | | 1) ( )( )( ) 2)( )( )( ) 3) 4) 0 33) If ax4 + bx3 + cx2 + dx + e = | | then e = 1) 2 2) 5 3) 0 4) 1 34) The value of | | 1) ( ) 2) ( ) 3) ( ) 4) ( ) 35) The value of | | 1) ( )( )( ) 2) ( )( )( ) 3) ( )( )( ) 4) ( )( )( ) 36) The value of | | 1) ( ) 2) ( ) ( ) ( ) 3) ( ) ( ) ( )
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