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 ant 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 nill-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 nill-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 . |