Properties of Determinants |
† The value of the determinant is not changed by changing the rows into columns and vice versa. |
† If 2 rows or columns of a determinant are interchanged, the sign of the determinant changes but numerical value remains the same. |
† If any line of a determinant is passed over p parallel lines, the resultant determinant is (-1)^{p}Δ. |
† If any 2 rows or 2 columns of a determinant are identical, the determinant vanishes. |
† The determinant that is left by canceling the row and column intersecting at a particular constituent is called the minor of the constituent & is denoted by the corresponding capital letter. |
† If the constituent of any row or column are multiplied with minors of the constituents of some other row or column and summed up with alternating plus and minus sign, the result is zero. |
† Co-factors are equal to minors in magnitude but may be of same or opposite sign according to the constituent whose co-factor is being calculated is brought to top left hand corner by even or odd movement of |
† If each constituent in any row or in any column is multiplied by the same factor, then the determinant is multiplied by that factor. |
† If each constituent in any row or column consists of 2 terms, then the determinant can be expressed as the sum of 2 determinants. |
† If to each constituent of a line of a determinant are added or subtracted the equi -multiples of the corresponding constituents of one or more lines, the determinant remains unaltered. |
† If rth line has been altered by means of sth & t th parallel lines,then sth & t th lines cannot themselves be altered by means of rth line, but we can use the rth line to alter all other parallel lines except the tth and sth line. |
† If the constituents of a determinant which involve x, are polynomials in x and if Δ = 0, when a is substituted for x, then x - a is a factor of the determinant. |
Properties of Matrices |
• A matrix of order n x m obtained from a matrix m x n by interchanging its rows and columns is called TRANSPOSE of original matrix. Transpose of transpose of a matrix coincides with the matrix. |
• Upper triangular Matrix-- A square matrix of order n x n is called upper triangular matrix if its elements a_{ij} = 0 for i > j for all i and j ranging from 1 to n. Thus an upper triangular matrix has non-zero elements only in the upper triangle above the principal diagonal whereas all the elements in the lower triangle are zero. |
• Lower triangular matrix--Similarly if all the elements in the upper triangle of a square matrix are zero, it is known as the lower triangular matrix. It has non-zero elements only in the lower triangle below the principal diagonal. |
• Diagonal Matrix-- A square matrx of order n which is both upper triangular and lower triangular is called diagonal matrix. A diagonal matrix is one all of whose elements are zero except those in the principal diagonal. |
• Scalar Matrix -- A diagonal matrix all the diagonal elements of which are equal to a scalar quantity, say λ, is called a scalar matrix. |
• Unit Matrix-- A scalar matrix with each diagonal element equal to unity is called a Unit Matrix. |
• Periodic Matrix-- A square matrix A for which A^{k+1} =A, is said to be a periodic matrix and the least positive value of k satisfying this relation is called the period of this periodicity. For exa, A is a matrix with a11=2,a12=-2,a13=-4 a21=-1,a22=3,a23=4,a31=1,a32=-2,a33=-3, then A*A = A. So here k=1 & hence A is a matrix of period 1. A matrix of period 1 is called idempotent matrix. If for a matrix A, there exists a positive integer p such that A^{p} = 0, , then A is called nil-potent matrix & p is called nill potency. Exa- A is a matrix with a11=1,a12=1,a13=3 a21=5,a22=2,a23=6,a31=-2,a32=-1,a33=-3, then A^{3} = 0 or null matrix. A is therefore a nil-potent matrix with the index of nill potency equal to 3. |
• Symmetric matrix-- A square matrix is said to be symmetric if it is unaltered by interchanging rows & columns, a_{ij} = a_{ji} |
• Anti-Symmetric matrix-- A square matrix is said to be anti-symmetric if its sign is changed by interchanging rows & columns,,i.e a_{ij} = -a_{ji} |
• Conjugate of a Matrix- When the elements of a matrix are complex, then a matrix whose elements are complex conjugates of A is called Conjugate of A. |
• Conjugate Transpose of A -- Conjugate of the transpose of a matrix A is called Conjugate Transpose of A & is denoted by A† |
• Hermitian Matrix : A square matrix conjugate transpose of which coincides with the matrix itself is called Hermitian matrix. A square matrix conjugate transpose of which coincides with the matrix itself but with negative sign is called Anti-Hermitian. Determinant of a Hermitian Matrix is real. |
• Unitary or Orthogonal Matrices :- A square matrix is said to be unitary if its inverse is equal to its conjugate transpose. A real unitary matrix is called Orthogonal matrix. |
• Adjoint of a Matrix--Adjoint or Adjugate of a matrix is the transpose of a matrix whose elements are cofactors of corresponding elements of the given matrix. |
•Soln of Cubic Equation : x^{3} +a2x^{2}+ax+a0=0 is (x-z1)(x^{2} +kx+l)=0 where k =a2+z1 ; l= -a0/z1 ; Once 1 of the roots are found out, the rest are easy. p=a1-(a2)^{2}/3 ; q=2(a2)^{3} / 27 - a2*a1/3 + a0 ; Δ= q^{2}/4 +p^{3}/27 ; If Δ < 0, there are 3 real roots ; if Δ =0 . one root is different & 2 other roots are same & if Δ > 0, there is only one real root. |
Trace of a matrix is invariant under similar transformation i.e. B =S^{-1}AS where B is a diagonal matrix . |