MATHEMATICS Matrix Algebra by Dr. Eman Saad & Dr. Shorouk Ossama

3.1 Matrix definition: In many applications, it is useful to be able to represent and manipulated data in tabular or array form. An array which obeys certain algebraic rules of operation is known as a matrix. Capital letters are usually used to denote matrices. Thus

A is a matrix with two rows and three columns 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 -2. 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. 2x3

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 m x n. A (1x1) matrix is simply a number: for example, [-3] = -3. Matrices which have either one row or column are known as vectors. Thus a = [ a1 a2 ……. an] are respectively row and column vectors

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.

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

Solve the equation A=B when Example Solve the equation A=B when and 2x3 2x3 Since and must have the same elements, if follows that And Hence, The solution is:

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

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 + 0 = 0 + A = A A – A = 0 0 – A = - A A 0 = 0; 0 A = 0

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 m x n matrices, then and

Then find A+B, B+A and A+2B. Example If Then find A+B, B+A and A+2B. (by Rule 2) and 3x2 3x2

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

SUMMARY Pages From 31 To 35: Matrix Algebra Sheet (3) (problem 1)

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