1. Matrices and Determinants. Matrix:-An arrangement numbers (real or complex
)in the form of rows and columns within the brackets is called a Matrix.
Matrices and Determinants Matrix:-An arrangement numbers (real or complex )in the form of rows and columns within the brackets is called a Matrix. The numbers that form a matrix called elements of the matrix. The matrices are denoted by capital letters If a matrix has , then A is called the matrix of order (read as ) A matrix is just a system of numbers. It has no numerical value. For example i)
is a matrix of order
ii) is a matrix of order Recall:- Types of matrices:-Row, Column, Rectangular, Square, Zero Or Null, Diagonal Elements of the matrix, Diagonal, Scalar, Unit or Identity Matrix. Operations on matrices: Equality of matrices.(same order and their corresponding elements are equal, then A=B) Scalar multiplication of matrices.( multiplying each element of A by a constant k ) Addition and subtraction of matrices.(same order and their corresponding elements are Added or subtracted) Multiplication of matrices.(number of columns in A=number of rows in B, then AB exists ) Transpose of a matrix.( interchanging the rows and columns of A, denoted by ) A square matrix A is called Symmetric matrix if A square matrix A is called skew- Symmetric If
Determinants: A real number value associated with a square matrix is called determinant. The value of determinant of order 2 is equal to the product of the elements along the principal diagonal minus the product of the second diagonal elements. Properties of Determinants: The value of a determinant remains unaltered if its rows are changed into columns and vice versa If any two rows or columns of a determinant are interchanged, the sign of the determinant is changed. If any two rows or columns of a determinant are identical, then value of the determinant is zero. If each element in a row or column of a determinant is multiplied by a constant, then value of the determinant is multiplied by the same constant. If all the elements of any row or column of a determinant is sum of the two terms, then the determinant can be expressed as sum the two determinants. The value of a determinant is not altered, if to the elements of any row or column the same multiples of the corresponding elements of any row or column are added. If A, B are square matrices of the same order ,then = . If
(product of the diagonal elements)
Adjoint of a matrix:-Recall minors and cofactors, singular ( singular matrix, Inverse of a matrix,
and non .
.
Characteristic equations of square matrices: Cayley-HamiltonTheorem: (Only Statement). ‘Every square matrix satisfies its own characteristic equation’. 1
QUESTIONS
1)
If
2)
If
3)
If A=
show that
4)
If A=
B=
5)
If A=
6)
If A is any square matrix then show that
7)
=
B=
find A and B.
find and .
[1m] [2m]
show that
[2m]
is a scalar matrix, find and .
i)
is a symmetric matrix
ii)
is a symmetric matrix
If A=
[1m]
then prove that
[1m]
[1m each]
i)
=
ii)
. [1m each]
8)
If
9)
If A=
, where I is the identity matrix, find x and y. B=
find
.
[2m] [1m]
10)
[1m]
11)
[1m]
12)
[1m]
13)
If
14)
Evaluate
find
[1m]
[1m]
2
15)
If
is one of the imaginary cube roots of unity, prove that
[1m]
16)
Show that
[3m]
17)
Show that
[3m]
18)
Show that
[4m]
19)
Show that
[4m]
20)
Show that
[4m]
21)
Show that
[4m]
22)
If [4 or 5m]
23)
Prove that
24)
Solve for
25)
If A
[4m]
=0
[3m]
then prove that [5m]
26)
Find the inverse of the matrix; i)
ii)
. 3
[5m]
27)
Solve by Cramer’s rule :
28)
Solve by Cramer’s rule:
[2m]
; 29)
Solve by matrix method:
30)
Solve by matrix method:
[5m] [3m]
[5m] 31)
Find the eigen values (characteristics roots) of the matrix:
32)
State Cayley-Hamilton Theorem. Find Eigen values and verify Cayley-Hamilton theorem for the matrix
33)
.
.
[4m]
State Cayley - Hamilton Theorem. Find
, if A=
using Cayley - Hamilton
Theorem. 34)
Find
[2m]
[4m]
, using Cayley-Hamilton theorem if
4
.
[4m]
