12 Matrices,representation of equations in matrix form and their solution.

Prof. Aslam Mahmood

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Matrix algebra is an important branch of mathematics dealing with the handling of the large body of data for its tabulation, representation and statistical analysis. It helps in representing large body of equations and their mathematical analysis in a fast and concise way. In this age of quantitative revolution, world is flooded with huge amount of data in every walk of the life and computers play an important role in analyzing the data in a much faster way. Most of theComputer software for multivariate analyses, therefore, require data input in matrix forms only.

 

Definition

 

Matrix is an arrangement of numbers in a rectangular form of rows and columns like the one given below.

 

2     4      6

[7    3      1]

3     1       9

 

A matrix could have several rows and columns. In the above case the matrix has three columns

5     2

 

and three rows. A matrix of three rows and two columns would look like: [8 2]similarly a

4    7

 

matrix with two rows and three columns would look like: [5  6  2] . A matrix which only one

1  3  7

 

column is known as a column vector like: [3] and similarly a matrix with only one row is known 8 as a row vector like; [12 5 11]. Number of rows and columns of a matrix are kwon as it dimensions and are written as mxn which means m-rows and n- columns. If the dimensions of a matrix are 3×3 it means 3 rows and 3 columns. Similarly the dimensions of a matrix with 4 rows and 7 columns will be written as 4×7.. A matrix which has equal number of rows and columns is known as square matrix, like 2×2, 3×3, 4×4 , 6×6 etc . If number of columns and rows of the matrix are not equal it is known as rectangular matrix. However a matrix with one column or one row only is known as vector. Dimensions of a four value column vector will be 4×1, as it will have 4 rows and only one column and a 5 values row vector will be 1×5 as it will have one row only but five columns. Each element of a matrix is known as an element and is denoted by aij,where I and j are known as subscript and represent the number of row and column respectively. The element a2x3 means the element corresponding to 2nd row and 3rd column. The elements of a square matrix corresponding to same row and column like a1x1, a2x2 , a3x3….. etc are know as principal diagonal elements.

 

There are certain types of matrices which have their equivalence in arithmetic. For example equivalence of zero in arithmetic is null matrix in matrix algebra:

 

Null Matrix

 

A null matrix is a square matrix whose all cell values are zero. Like a 3x3null matrix would 0 0 0

be:[0  0  0]

0   0   0

 

Identity or unit matrix

 

Similarly equivalence of Unity or one in matrix is an Identity matrix which is a square matrix whose all the diagonal elements are one and the off diagonal elementsare all zeros. A 2×2

 

identity matrix would be: [1  1  0  0

0]and a 3×3 Identity matrix would be:[0  1  0]. An identity

0  1  0  0  1

 

matrix is denoted I.

 

Diagonal Matrix

 

A matrix whose all off diagonal elements are all zeros but diagonal elements are other than

2  0  0

zeros is known as diagonal matrix. For example the matrix [0  3  0] is a diagonal matrix.

0   0  9

 

Symmetric Matrix

 

A square matrix whose element aijis equal to element aji is known as symmetric matrix. The

2  7  3

matrix [7  3  1] is a symmetric matrix as among all of its off diagonal elements aij = aji. One

3  1  9

 

can verify that element a2x3 = a3x2 =1 , element a1x3= a3x1 = 3 and element a1x2 = a2x1=7. Likewise [26 67]Is a 2×2 symmetric matrix. Similarly we can have symmetric matrix ( square)of any other dimensions also.

 

Transpose of a matrix

 

If the rows and columns of a matrix A are interchanged, the new matrix is known as transpose

5  2

of A and is represented by a dash on it, for example if we have A = [8 2] transpose of A′ =

 

4  7

[3  2  7]

1  2  3

 

 

number of columns and number of columns become number of rows. In the present case A is a 3 x 2 matrix but A’ is a 2 x 3 matrix. A row vector after transpose will become a column vector. A symmetric matrix, however, remains unchanged after transpose. Thus another definition of a square symmetric matrix is that if A’ = A , the matrix is said to be symmetrical.

 

 

Scalar Matrix

 

Any square matrix whose all the diagonal elements are equal and all the off diagonal elements 2 0 0 are zero is known as diagonal matrix. For example a matrices[80  08] and [0  3  0] 0  0  9 are scalar matrices of the order of 2×2 and 3×3.

 

Like simple algebra all the four arithmetic operations are possible in matrix also. In matrix algebra these operations of different sets of numbers are performed simultaneously.

 

Matrix Addition and Subtraction

Two matrices of identical dimensions can be added or subtracted by adding or subtracting their values of the corresponding elements. It is to be noted that if the dimensions of any two matrices are not equal they cannot be added or subtracted.

 

Example

 

 

One can verify that the value of the elements of the last matrix giving the sum is the sum of the values of the corresponding elements of the given matrices. For example 3 + 1 = 4 is the sumof the values of first row and first column of two matrices.

 

Similarly if we have to subtract:

 


The resultant matrix will have the difference of the corresponding values of the two matrices as show above.

 

 

 

Matrix Multiplication

 

Multiplication of two matrices also involves multiplication of their elements but in a different way. It takes the elements of the first row of the first matrix and multiply the corresponding elements of the first column of the second matrix to get a1x1 of the product matrix. Thus number of columns of first matrix ( number of elements in it ) should be equal to number of rows ( number of elements in the columns )of the second matrix should be equal. Similarly we repeat the process by taking the second column of the second matrix we get the value of a1x2 of the product matrix. By changing the columns of the second matrix, retaining the first column of the first matrix we get different values of the first row of the product matrix. We repeat the process by taking the second row of the first matrix and get different values for the second row of the product matrix . At the end we get the values of the product matrix with rows equal to the number of rows of the first matrix and columns equal to number of columns of the second matrix. So if the dimensions of the two matrices are 3×4 and 4×5, the dimension of the product matrix will be 3×5. Note that number of columns of the first matrix is 4 and number of columns of the second matrix is also 4 The two matrices can be multiplied and the dimensions of the product matrix will be 3×5 i.e. number of rows of first matrix 3 and number of columns of the second matrix 5.

 

Example If,

 

 

We note that the two matrices can be multiplied as they have required dimensions but the two product are not same i.e. A x B ≠ B x A. Also note that when an Identity or unit matrix is multiplied to any matrix confirming the dimensions, the product is the matrix itself i.e. I x A = A and A x I =A. Thus the commutative law will hold good only if one of the factor matrix is identity matrix. i.e. I xA = I x A

First multiplier is known as pre factor and the second is known as post factor.

 

Example

 

If the dimensions of the two matrices are 2 x 3 and 3 x 3 like:

 

 


The dimensions of the product A x B will be ( 2 x 3 and 3 x 3 ) 2 x 3. We note that B x A is not possible as the dimensions of the pre factor B in this case are such that it can not be multiplied with the post factor A. 

 

Scalar Multiplication

 

 

 

Minor and cofactors of a square matrix

 

 

 

Determinant of a square matrix

 

Every square matrix has a value which is known its determinant and is denoted by two vertical lines on either side of it. The determinant of a square matrix A will be denoted by I A I. Determinant of a 2 x 2 square matrix is worked out by taking the difference of the product of diagonal terms and the product of remaining off diagonal terms. For a three by three

 

 

 

A square matrix is known as singular matrix if its determinant is found to be zero.

 

Properties of determinants.

 

In any square matrix following are the properties of the determinants:

 

  1. Interchanging of rows and columns will not change the value of the determinant.
  2. In any row or column if all the values are zeros, the value of the determinant will be zero.
  3. Interchanging any two rows or columns will reverse the sign of the value of the determinant.
  4. If any two rows or column of a matrix are same the value of the determinant will be zero.
  5. If the values of any row or column are multiplied by a constant the value of the determinant will also get multiplied by that constant.
  6. If to any row or column any other row or column is added after multiplying by a constant, the value of the determinant will not change.

 

Adjoint of a square matrix.

 

Transpose of a matrix formed by the cofactors of its each element is known as its adjoint. For example if we have a matrix A =

 

 

Inverse of a matrix

 

Inverse of a matrix is equivalent to division in arithmetic but in a very indirect way. In arithmetic if a number A is divided by another number B , we get AxB=1 and B is said to be the inverse of A and A inverse is B. A inverse is denoted by A-1.

 

 

Computation of the Inverse of a symmetric matrix.

 

The inverse of a non singular symmetric matrix A is worked out by using the relationshipA-1 =IAI1 ∗ Adjoint A, provided IAI is not zero i.e. A is a non-singular matrix.

 

 

Note that as the dimensions of matrix increases computation of its inverse becomes more and more difficult and we have to take the help of computers and statistical packages for its calculations.

 

 

 

Simultaneous Equations and Their Solution.

 

Biggest advantage of matrix methods is that it can simultaneously handle a large body of information. Taking advantage of this property we can represent and solve a large number of equations in a concise and efficient manner. For example if we have a set of three simultaneous linear equations in three variables X1, X2 and X3 as given below:

 

 

This system of three simultaneous linear equations can also be written in matrix form as:

 

 

     These equation can be further written in a concise form as :

 

A X = C

 

where A is a 3×3 square matrix and X an C are column vectors ( matrix of the order of 3 x 1). Solution of these equations is possible by pre-multiplying both the sides by A-1

 

which will give(A-1A)X =A-1C or :

 

X = A-1 C

 

Note that (A-1A)= I multiplication of I with X will not change X .The above equation will hold good for any number of variables with the condition that number of independent equations should be equal to the number of variables to give us a non singular symmetric matrix of A. In case the number of equations are not independent, matrix A will become singular and inverse of matrix A will not exit.

   

Example

 

Solve the following equations.

 

 

 

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References

  • Aslam Mahmood (1998) Statistical Methods in Geographical Studies, Rajesh Publications New Delhi.
  • Hadley G. (1962) . Linear Algebra, Addison-Wesley Publishing Co. Inc.
  • Sydsaeter K. and Hammond P. J. (2012). Pearsons Educations, Inc.