Matrices

King Saud University

• If m and n are positive integers, then an m n matrix is a rectangular array in which each entry aij of the matrix is a number. The matrix has m rows and n columns.

11 12 13 1

21 22 23 2

31 32 33 3

1 2 3

n

n

n

m m m mn

a a a a

a a a a

a a a a

a a a a

Terminology

• A real matrix is a matrix all of whose entries are real numbers.

• i (j) is called the row (column) subscript.

• An mn matrix is said to be of size (or dimension) mn.

• If m=n the matrix is square of order n.

• The aii’s are the diagonal entries.

• Given a system of equations we can talk about its coefficient matrix and its augmented matrix.

• These are really just shorthand ways of expressing the information in the system.

• To solve the system we can now use row operations instead of equation operations to put the augmented matrix in row echelon form.

Elementary Row Operations

1. Interchange two rows.

2. Multiply a row by a nonzero constant.

3. Add a multiple of a row to another row.

• Two matrices are said to be row equivalent if one can be obtained from the other using elementary row operations.

• A matrix is in row-echelon form if:– All rows consisting entirely of zeros are at the

bottom.– In each row that is not all zeros the first entry is

a 1.– In two successive nonzero rows, the leading 1

in the higher row is further left than the leading 1 in the lower row.

Gaussian Elimination with Matrices

1. Write the augmented matrix of the system.

2. Use elementary row operations to find a row equivalent matrix in row-echelon form.

3. Write the system of equations corresponding to the matrix in row-echelon form.

4. Use back-substitution to find the solutions to this system.

Gauss Jordan Elimination

• In Gauss-Jordan elimination, we continue the reduction of the augmented matrix until we get a row equivalent matrix in reduced row-echelon form. (r-e form where every column with a leading 1 has rest zeros)

1 0 0

0 1 0

0 0 1

a

b

c

Homogeneous Systems

• A system of linear equations in which all of the constant terms is zero is called homogeneous.

• All homogeneous systems have the solutions where all variables are set to zero. This is called the trivial solution.