Mathematics Medicine

What is meant by a matrix A matrix is a set of numbers arranged in the form of a rectangle and enclosed in curved brackets. The plural of matrix is matrices. Each number in a matrix is known as an element. To refer to a particular matrix we label it with a capital letter, so that we could write We refer to the size of a matrix by giving its number of rows and number of columns, in that order. So matrix A above has n rows and m columns - we say it is a ’n by m' matrix, and write this as ‘n x m'.

Examples 2 x 3 - 'two by three' matrix 3 x 1 – ‘three by one' matrix 3 x 3 – ‘three by three' matrix

Special matrices Square matrix A diagonal matrix Identity matrix

Square matrix A square matrix has the same number of rows as columns. are both square matrices. and

Diagonal matrix A diagonal matrix is a square matrix in which all the elements are except those on the diagonal from the top left to the bottom right. The diagonal is called the leading diagonal. On the leading diagonal the elements can take any value including zero. are all diagonal matrices.

Identity matrix An identity matrix is a diagonal matrix, all the diagonal entries of which are equal to 1 are all identity matrices

Addition, subtraction and multiplication of matrices Matrices can be added, subtracted multiplied. However, they can never be divided.

Addition Two matrices having the same size can be added by simply adding the corresponding elements.

Subtraction Two matrices having the same size can be subtracted by simply subtracting the corresponding elements.

Different size The matrices and can be neither added to nor subtracted from one another because they have different sizes

Multiplying a matrix by a number A matrix is multiplied by a number by multiplying each element by ti number. and

Multiplying two matrices together Two matrices can only be multiplied together if the number of columns in the first is the same as the number of rows in the second. The product of two such matrices is a matrix that has the same number of rows as the first matrix and the same number of columns as the second. In symbols, this states that if A has size pxq and В has size qxs, then AB has size pxs. 2 x 33 x 2 2 x 2

Example 2 x 22 x 1

Example

Note that in this example the result is a 1x1 matrix, that is, a single number

Example

Note that the effect of multiplying the matrix Х by the identity matrix is to leave X unchanged. This is a very important and useful property of identity matrices, which we shall require later. It should be remembered.

The inverse of a 2  2 matrix An important matrix that is related to A is known as the inverse of A and is given the symbol A -1. A -1 can be found from the following formula: Ifthen This formula states that: the elements on the leading diagonal are interchanged the remaining elements change sign the resulting matrix is multiplied by The inverse matrix has the property that:

Example Find the inverse of the matrix Using the formula for the inverse we find

Determinant If The quantity ad - bc in the formula for the inverse is known as the determinant of the matrix A. We often write it as |A|, the vertical bars indicating that we mean the determinant of A and not the matrix itself.

Example Find the value of the determinant Solution. The determinant is given by (8)(2) - (7)(9) = = -47.

Singular matrix On some occasions a matrix may be such that ad—bc=0, that is, its determinant is zero. Such a matrix is called singular. When a matrix is singular it cannot have an inverse. This is because it is impossible to evaluate the quantity 1/(ad— bc), which appears in the formula for the inverse; the quantity 1/0 has no meaning in mathematics.

Application of matrices to solving simultaneous equations Matrices can be used to solve simultaneous equations. Suppose we wish to solve the simultaneous equations 3x + 2y = -3 5x + 3y = -4 First of all we rewrite them using matrices as we have AX=B.Writingand Note that we are trying to find x and y; in other words we must find X.

Now, if AX=B then, provided A -1 exists, we multiply both sides by A -1 to obtain A -1 AX= A -1 B But A -1 A=I, so this becomes IX= A -1 B However, multiplying X by the identity matrix leaves X unaltered, so we can write X= A -1 B In other words if we multiply В by the inverse of A we will have X as required.

Now, given Then Finally, Therefore the solution of the simultaneous equations is x = 1 and у=-3