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Matrices

matrix

Each symbol such as a11, a2n, or amn is called an element of the matrix.

The square bracket enclosing the array signifies that it is a matrix.

The two straight lines enclosing a square array signify that to be a determinant.

The matrix A contains m rows and n columns.

A is called an m by n matrix, or A is said to be of order “m×n”.

2 by 3 matrix implying that there are two rows and three columns

row matrix-

having n element arranged in a single row

column matrix-

having n element arranged in a single column

unit column matrix

square matrix-

having the number of rows of elements equivalent to the number of columns of elements.

diagonal matrix-

all the elements except those in the diagonal from the top left-hand corner to the bottom right-hand corner are zero.

unit matrix-

a diagonal matrix in which the diagonal elements are all unity.

unit matrix of order three

null matrix-

all elements in matrix are zero, and is denoted by 0.

1.This operation can only be carried out on matrices of the same order.

2.The sum of two matrices is obtained by adding together the corresponding

elements of each matrix.

Ex:

Note:the sum or difference does not exist if the matrices are of different order

Scalar Multiplication

If any matrix is multiplied by a number, the elements of the matrix so formed are the products of the number and the corresponding elements

of the original matrix.

Matrix Multiplication

The product of two matrices will exist only when the number of columns of the first matrix is equal to the number of rows in the second matrix. When this condition is satisfied the two matrices are said to be

“conformable” and they yield a product.

If the matrix A is of order (m×n) and the matrix B is of order (n×m) the product C of AB=C is a matrix of order (m×n).(n×m)=(m×m). On the other hand, the product D of BA=D is matrix of order (n×m).(m×n)=(n×n).

(i) C is obtained by pre-multiplying B by A.

(ii) C is obtained by post-multiplying A by B.

Matrix Multiplication

In the exceptional case when the products of two square matrices are equal, i.e. AB=BA, the matrices are said to commute, or be commutable.

Note:

The determinant of the product of two square matrices is equal to the product of their determinants.

Even though in general ABBA

Note:

If A is a square matrix and , the matrix is called a “singular matrix”. On the other hand, if , the matrix is “non-singular”.

If the rows of an (n×m) matrix are written in the form of columns, a new matrix of order (m×n) will be formed. This new matrix is called

the “transpose” of the original matrix.

Note:

A matrix and its transpose are always conformable for multiplication and

if AA’=I, the matrix A is said to be “orthogonal”.

Note:

The transpose of the product of two matrices is equal to the product of

their transposes taken in the reverse order.

If aij is any element of a sqaure matrix of order (nn), and the cofactor of aij in the determinant is Aij, the transpose of the matrix whose elements

are made up of all the Aijs is called the “adjoint of A”, and is written as

Consider a square matrix A to be non-singular and of order (nn)

Define a new matrix

x1=5.704, x2=0.261, x3=1.452, x4=1.313

Powers of Matrices

Matrix Polynomials

X is a square matrix

Differentiation

Integration

A -matrix is a square matrix in which the elements are function of a scalar parameter .

when =1 the determinant of the matrix () is zero, and hence this matrix is singular. However, its minors of order two are non-

vanishing, therefore the -matrix is of rank 2.

Let A be a square matrix of order n whose elements are all constants, and let x be a column whose elements are related by the equation

where  is a scalar coefficient.

where (A-I) is a square -matrix of order n.

characteristic equation of matrix A

Roots p are called the “latent roots” or the “eigenvalues” of A.

Note: A and its transpose have the same latent roots.

The vector corresponding to the latent root is called a “latent column vector”. Let k1, k2, k3, …,kn be the latent column vectors corresponding to the latent roots of A, and K the square matrix comprising all the column

vectors. K will be of order n and non-singular.

Showing that the matrix A can be reduced to what is known as “diagonal canonical form”

Evaluate the latent roots and reduce the following matrix to diagonal form

Any square matrix satisfies its own characteristic equation.

The characteristic equation of the above matrix is

Calculate A4 if

The characteristic equation of the matrix A is

Replacing  by A and rearranging gives

Multiplying by A gives

Multiplying by A gives

Consider the matrix

Find

(a) A1/2=?

(b) sinA=?

(c) exp(A2)=?

System of first order linear ordinary differential equation

n=1

n>1

Solve

Solve

Consider system of first order non-linear ordinary differential equation

Let

Solve

System of second order homogeneous ordinary differential equation

Let

Solve

Let