*Matrices and Determinants*

*On this page:*

*Introduction and Examples*

*Matrix Addition and Subtraction*

*Matrix Multiplication*

*The Transpose of a Matrix*

*The Determinant of a Matrix*

The Inverse of Matrix

Systems of Linear Equations

The Inverse Matrix Method

Cramer’s Rule

Introduction and Examples

DEFINITION: A matrix is defined as an ordered rectangular array of numbers. They can be used to represent systems of linear equations, as will be explained below.

*Here are a couple of examples of different types of matrices:*

*Symmetric Diagonal Upper Triangular Lower Triangular Zero Identity*

*Symmetric Matix Diagonal Matrix Upper Triangular Matix Lower Triangular Matix Zero Matix Identity Matix*

*And a fully expanded m×n matrix A, would look like this:*

*n×n matrix*

*… or in a more compact form: m×n simplified*

*Top*

*Matrix Addition and Subtraction*

*DEFINITION: Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the same number of rows and columns. Take:*

*matrices A&B*

*Addition*

*If A and B above are matrices of the same type then the sum is found by adding the corresponding elements aij + bij .*

*Here is an example of adding A and B together.*

*Sum of matrices A&B*

*Subtraction*

*If A and B are matrices of the same type then the subtraction is found by subtracting the corresponding elements aij − bij.*

*Here is an example of subtracting matrices.*

*Subtraction of A&B*

*Now, try adding and subtracting your own matrices.*

*Addition/subtraction Top*

*Matrix Multiplication*

*DEFINITION: When the number of columns of the first matrix is the same as the number of rows in the second matrix then matrix multiplication can be performed.*

*Here is an example of matrix multiplication for two 2×2 matrices.*

*Matrix multiplication 2×2*

*Here is an example of matrix multiplication for two 3×3 matrices.*

*Matrix multiplication 3×3*

*Now lets look at the n×n matrix case, Where A has dimensions m×n, B has dimensions n×p. Then the product of A and B is the matrix C, which has dimensions m×p. The ijth element of matrix C is found by multiplying the entries of the ith row of A with the corresponding entries in the jth column of B and summing the n terms. The elements of C are:*

*Matrix multiplication for n×n*

*Note: That A×B is not the same as B×A*

*Now, try multiplying your own matrices.*

*Matrix multiplication Top*

*Transpose of Matrices*

*DEFINITION: The transpose of a matrix is found by exchanging rows for columns i.e. Matrix A = (aij) and the transpose of A is:*

*AT = (aji) where j is the column number and i is the row number of matrix A.*

*For example, the transpose of a matrix would be:*

*Transpose of matrix*

*In the case of a square matrix (m = n), the transpose can be used to check if a matrix is symmetric. For a symmetric matrix A = AT.*

*Symmetric matrix*

*Now try an example.*

*Transpose of a matrix Top*

*The Determinant of a Matrix*

*DEFINITION: Determinants play an important role in finding the inverse of a matrix and also in solving systems of linear equations. In the following we assume we have a square matrix (m = n). The determinant of a matrix A will be denoted by det(A) or |A|. Firstly the determinant of a 2×2 and 3×3 matrix will be introduced, then the n×n case will be shown.*

*Determinant of a 2×2 matrix*

*Assuming A is an arbitrary 2×2 matrix A, where the elements are given by:*

*Matrix A*

*then the determinant of a this matrix is as follows:*

*Det A*

*Now try an example of finding the determinant of a 2×2 matrix yourself.*

*Determinant of 2×2*

*Determinant of a 3×3 matrix*

*The determinant of a 3×3 matrix is a little more tricky and is found as follows (for this case assume A is an arbitrary 3×3 matrix A, where the elements are given below).*

*Matrix A*

*then the determinant of a this matrix is as follows:*

*Det of A*

*Now try an example of finding the determinant of a 3×3 matrix yourself.*

*Determinant of 3×3*

*Determinant of a n×n matrix*

*For the general case, where A is an n×n matrix the determinant is given by:*

*Matrix A n×n*

*Where the coefficients αij are given by the relation:*

*alpha coefficient*

*where βij is the determinant of the (n-1) × (n-1) matrix that is obtained by deleting row i and column j. This coefficient αij is also called the cofactor of aij.*

*Top*

*The Inverse of a Matrix*

*DEFINITION: Assuming we have a square matrix A, which is non-singular (i.e. det(A) does not equal zero), then there exists an n×n matrix A-1 which is called the inverse of A, such that this property holds:*

*AA-1 = A-1A = I, where I is the identity matrix.*

*The inverse of a 2×2 matrix*

*Take for example a arbitury 2×2 Matrix A whose determinant (ad − bc) is not equal to zero.*

*2×2 matrix*

*where a,b,c,d are numbers, The inverse is:*

*Inverse of 2×2*

*Now try finding the inverse of your own 2×2 matrices.*

*Inverse of 2×2*

*The inverse of a n×n matrix*

*The inverse of a general n×n matrix A can be found by using the following equation.*

*Inverse*

*Where the adj(A) denotes the adjoint (or adjugate) of a matrix. It can be calculated by the following method:*

*Given the n×n matrix A, define*

*B = bij*

*to be the matrix whose coefficients are found by taking the determinant of the (n-1) × (n-1) matrix obtained by deleting the ith row and jth column of A. The terms of B (i.e. B = bij) are known as the cofactors of A.*

*Define the matrix C, where*

*cij = (−1)i+j bij.*

*The transpose of C (i.e. CT) is called the adjoint of matrix A.*

*Lastly to find the inverse of A divide the matrix CT by the determinant of A to give its inverse.*

*Now test this method with finding the inverse of your own 3×3 matrices.*

*Inverse of 3×3 Top*

*Solving Systems of Equations using Matrices*

*DEFINITION: A system of linear equations is a set of equations with n equations and n unknowns, is of the form of*

*n×n Systems of equations*

*The unknowns are denoted by x1, x2, …, xn and the coefficients (a and b above) are assumed to be given. In matrix form the system of equations above can be written as:*

*n×n Systems of equations*

*A simplified way of writing above is like this: Ax = b*

*Now, try putting your own equations into matrix form.*

*Putting equations into matrices*

*After looking at this we will now look at two methods used to solve matrices. These are:*

*Inverse Matrix Method*

*Cramer’s Rule*

*Top*

*Inverse Matrix Method*

*DEFINITION: The inverse matrix method uses the inverse of a matrix to help solve a system of equations, such like the above Ax = b. By pre-multiplying both sides of this equation by A-1 gives:*

*Ax=b derivation*

*or alternatively*

*Ax=b derivation*

*So by calculating the inverse of the matrix and multiplying this by the vector b we can find the solution to the system of equations directly. And from earlier we found that the inverse is given by*

*Inverse*

*From the above it is clear that the existence of a solution depends on the value of the determinant of A. There are three cases:*

*If the det(A) does not equal zero then solutions exist using Ax=b derivation*

*If the det(A) is zero and b=0 then the solution will be not be unique or does not exist.*

*If the det(A) is zero and b=0 then the solution can be x = 0 but as with 2. is not unique or does not exist.*

*Looking at two equations we might have that*

*Inverse*

*Written in matrix form would look like*

*Inverse*

*and by rearranging we would get that the solution would look like*

*Inverse*

*Now try solving your own two equations with two unknowns.*

*Inverse Method 2×2*

*Similarly for three simultaneous equations we would have:*

*Inverse*

*Written in matrix form would look like*

*Inverse*

*and by rearranging we would get that the solution would look like*

*Inverse*

*Now try solving your own three equations with three unknowns.*

*Inverse Method 3×3 Top*

*Cramer’s Rule*

*DEFINITION: Cramer’s rule uses a method of determinants to solve systems of equations. Starting with equation below,*

*n×n Systems of equations*

*The first term x1 above can be found by replacing the first column of A by b×n. Doing this we obtain:*

*n×n Systems of equations*

*Similarly for the general case for solving xr we replace the rth column of A by b×n and expand the determinant. This method of using determinants can be applied to solve systems of linear equations. We will illustrate this for solving two simultaneous equations in x and y and three equations with 3 unknowns x, y and z.*

*Two simultaneous equations in x and y*

*2×2 equations*

*To solve use the following:*

*or simplified:*

*Now try solving two of your own equations.*

*Cramers 2×2*

*Three simultaneous equations in x, y and z*

*ax + by + cz = p*

*dx + ey + fz = q*

*gx + hy + iz = r*

*To solve use the following:*

*Now try solving your own three equations.*