PROGRAMME 5

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Matrices

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PROGRAMME 5

MATRICES

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Matrices – definitions

A matrix is a set of real or complex numbers (called elements) arranged in rows and columns to form a rectangular array.

A matrix having m rows and n columns is called an m×n matrix.

For example:

is a 2 × 3 matrix.

Programme 5: Matrices

Matrices – definitions

Row matrix

A row matrix consists of a single row. For example:

Column matrix

A column matrix consists of a single column. For example:

Programme 5: Matrices

Matrices – definitions

Double suffix notation

Each element of a matrix has its own address denoted by double suffices, the first indicating the row and the second indicating the column. For example, the elements of 3 × 4 matrix can be written as:

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Matrix notation

Where there is no ambiguity a matrix can be represented by a single general element in brackets or by a capital letter in bold type.

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Equal matrices

Two matrices are equal if corresponding elements throughout are equal.

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Two matrices are added (or subtracted) by adding (or subtracting) corresponding elements. For example:

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Multiplication of matrices

Scalar multiplication

Multiplication if two matrices

Programme 5: Matrices

Multiplication of matrices

Scalar multiplication

To multiply a matrix by a single number (a scalar), each individual element of the matrix is multiplied by that number. For example:

That is:

Programme 5: Matrices

Multiplication of matrices

Multiplication if two matrices

Two matrices can only be multiplied when the number of columns in the first matrix equals the number of rows in the second matrix.

The ijth element of the product matrix is obtained by multiplying each element in the ith row of the first matrix by the corresponding element in the jth column of the second matrix.

For example:

Programme 5: Matrices

Multiplication of matrices

Multiplication if two matrices

Programme 5: Matrices

Multiplication of matrices

Multiplication if two matrices

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Transpose of a matrix

If a new matrix is formed by interchanging rows and columns the new matrix is called the transpose of the original matrix. For example, if:

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Special matrices

Square matrix

Diagonal matrix

Unit matrix

Null matrix

Programme 5: Matrices

Special matrices

Square matrix

A square matrix is of order m×m.

A square matrix is symmetric if . For example:

A square matrix is skew-symmetric if . For example:

Programme 5: Matrices

Special matrices

Diagonal matrix

A diagonal matrix is a square matrix with all elements zero except those on the leading diagonal. For example:

Programme 5: Matrices

Special matrices

Unit matrix

A unit matrix is a diagonal matrix with all elements on the leading diagonal being equal to unity. For example:

The product of matrix A and the unit matrix is the matrix A:

Programme 5: Matrices

Special matrices

Null matrix

A null matrix is one whose elements are all zero.

Notice that

But that if we cannot deduce that or

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Determinant of a square matrix

Singular matrix

Cofactors

Programme 5: Matrices

Determinant of a square matrix

Singular matrix

Every square matrix has its associated determinant. For example, the determinant of

The determinant of a matrix is equal to the determinant of its transpose.

A matrix whose determinant is zero is called a singular matrix.

Programme 5: Matrices

Determinant of a square matrix

Cofactors

Each element of a square matrix has a minor which is the value of the determinant obtained from the matrix after eliminating the ith row and jth column to which the element is common.

The cofactor of element is then given as the minor of multiplied by

Programme 5: Matrices

Determinant of a square matrix

Let square matrix C be constructed from the square matrix A where the elements of C are the respective cofactors of the elements of A so that if:

Then the transpose of C is called the adjoint of A, denoted by adjA.

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Inverse of a square matrix

If each element of the adjoint of a square matrix A is divided by the determinant of A then the resulting matrix is called the inverse of A, denoted by A-1.

Note: if detA = 0 then the inverse does not exist

Programme 5: Matrices

Inverse of a square matrix

Product of a square matrix and its inverse

The product of a square matrix and its inverse is the unit matrix:

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Solution set of linear equations

The set of n simultaneous linear equations in n unknowns

can be written in matrix form as:

Programme 5: Matrices

Solution set of linear equations

Since:

The solution is then:

Programme 5: Matrices

Solution set of linear equations

Gaussian elimination method for solving a set of linear equations

Given:

Create the augmented matrix B, where:

Programme 5: Matrices

Solution set of linear equations

Gaussian elimination method for solving a set of linear equations

Eliminate the elements other than a11 from the first column by subtracting a21/a11 times the first row from the second row, a31/a11 times the first row from the third row, etc. This gives a new matrix of the form:

Programme 5: Matrices

Solution set of linear equations

Gaussian elimination method for solving a set of linear equations

This process is repeated to eliminate the ci2 from the third and subsequent rows until a matrix of the following form is arrived at:

Programme 5: Matrices

Solution set of linear equations

Gaussian elimination method for solving a set of linear equations

From this augmented matrix we revert to the product:

Programme 5: Matrices

Solution set of linear equations

Gaussian elimination method for solving a set of linear equations

From this product the solution is derived by working backwards from the bottom starting with:

Programme 5: Matrices

Matrices – definitions

Matrix notation

Equal matrices

Multiplication of matrices

Transpose of a matrix

Special matrices

Determinant of a square matrix

Inverse of a square matrix

Solution set of linear equations

Eigenvalues and eigenvectors

Programme 5: Matrices

Eigenvalues and eigenvectors

Equations of the form:

where A is a square matrix and  is a number (scalar) have non-trivial solutions (x0) for x called eigenvectors or characteristic vectors of A. The corresponding values of  are called eigenvalues, characteristic values or latent roots of A.

Programme 5: Matrices

Eigenvalues and eigenvectors

Expressed as a set of separate equations:

That is:

Programme 5: Matrices

Eigenvalues and eigenvectors

These can be rewritten as:

That is:

Which means that, for non-trivial solutions:

Programme 5: Matrices

Eigenvalues and eigenvectors

Eigenvalues

To find the eigenvalues of:

solve the characteristic equation:

That is:

This gives eigenvalues

Programme 5: Matrices

Eigenvalues and eigenvectors

Eigenvectors

To find the eigenvectors of solve the equation

For the eigenvalues  = 1 and  = 5

Programme 5: Matrices

Learning outcomes

• Define a matrix
• Understand what is meant by the equality of two matrices
• Add and subtract two matrices
• Multiply a matrix by a scalar and multiply two matrices together
• Obtain the transpose of a matrix
• Recognize the special types of matrix
• Obtain the determinant, cofactors and adjoint of a square matrix
• Obtain the inverse of a non-singular matrix
• Use matrices to solve a set of linear equations using inverse matrices
• Use the Gaussian elimination method to solve a set of linear equations
• Evaluate eigenvalues and eigenvectors