CALCULUS – IIIMatrix Operation

by

Dr. Eman Saad & Dr. Shorouk Ossama

References

Robert E. Moyer, Ph.D. Frank Ayers, Jr., Ph.D, Trigonometry, 4th Edition,2009.

3.1 Matrix definition:

• In many applications, it is useful to be able to represent and

manipulated data in array form. An array which obeys certain

algebraic rules of operation is known as a matrix. Capital letters are

usually used to denote matrices.

A is a matrix with two rows and three columns. The individual

terms are known as elements: the elements in the second

row and third column is -4. This matrix is said to be order 2×3,

or a 2×3 matrix. A general m×n matrix, one with m rows and n

columns, can be represented by the notation.

Where aij is the element in the ith row and jth column of A ; or by :

Or simply,

For brevity, if it is clear in context that the matrix is mxn.

A (1x1) matrix is simply a number: for example, [-3] = -3.

Matrices which have either one row or column are known as

vectors. Thus

are respectively row

and column vectors

a = [ a1 a2 ……. an]

•A matrix in which the number of rows equals the number of

columns is called a Square Matrix of Order n, if then the

matrix is said to be rectangular and the shaded elements a11,

a22, ….., ann are said the Main Diagonal of A.

3.2 Rules of matrix algebra:

We need to define consistent algebraic rules for

manipulating matrices, such concepts as addition,

multiplication, etc. As we shall see, these rules have their

origins in the representation of linear equations and linear

transformations, but for the present we simply state them

as a list of rules.

1. Equality: Two matrices are defined to be equal if they

have the same size and their corresponding elements are.

Then A = B, if and only if a=e, b=f, c=g, d=h, In general, if: A = [aij], B = [bij] are both mxn matrices,

Then A=B if and only if aij = bij for i= 1,2,….m and j=1,2,…m.

and

Example

Solve the equation A=B when

and

Since and must have the same elements, if follows that

And Hence, The solution is:

2. Multiplication by a constant: Let K be a constant

or scalar, By the product KA we mean the matrix in which

every element of A is multiplied by K. Thus, if :

And k= 10, then

Equally, we can 'factorize' a matrix. Thus

3. Zero matrix: Any matrix in which every element is zero

is called a zero or null matrix. If A is a zero matrix, we can

simply write A = 0

Assuming that sizes of the matrices are such that the

indicated operations can be performed, the following rules

of matrix arithmetic are valid.(a) A + 0 = 0 + A = A(b) A – A = 0(c) 0 – A = - A(d) A 0 = 0; 0 A = 0

4. Matrix sums and differences:

The sum of two matrices A and B has meaning only if A and B

are of the same order, in which case A+B is defined as

the matrix C, whose elements are the sums of the

corresponding elements in A and B .

write C = A + B Thus, if

are both mxn matrices, then

and

Example If

Then find A+B, B+A and A+2B.

We have (by Rule 2).

and

Also

As the second sum suggests, the commutative property of the

real numbers, namely aij+bij = bij+aij implies the commutative

property of matrix addition, that is A+B = B+A.

The difference of two matrices is written as which is

interpreted as A - B, A + (-1) B, using Rule 2 for the

multiplication of B by the number -1 , and then Rule 4 for the

sum of A and (-1) B. In practice, we simply take the difference

of corresponding elements.

Example

Find A - B, 2A - 3B if: and

The rules of arithmetic as applied to the elements of matrices

lead to the following results for matrices for which addition

can be defined:

In other words, the order of addition of matrices is immaterial.

Problems:

Find B – C, 2B – 3C, C + B if:

and

Thanks