ib maths sl matrices

A matrix is an ordered set of numbers listed in rectangular form

Matrix A has 2 rows and 3 columns. We say it is a 2x3 matrix.

order 2x3

B is a row matrix.

C is a column matrix.

This is the 3x3 zero-matrix.

I is the 3x3 Identity matrix.

and are opposite matrices.

We can sum matrices of the same order.

Multiplication of a matrix by a scalar

=3

To multiply matrices, we multiply rows into columns:

2 x 3 3x 2

=

2x 2

=

What special name has the answer?

Determinant of a matrixDeterminant of a matrix is a number calculated from the elements of the matrix.

A =

det A=

Determinant of a 3x3 matrix

Find the determinant of:

If P= , find the value of x for which

|P| = 0.

If the determinant of a matrix is zero , the matrix is called a singular matrix

Inverse of a matrix

Only square matrices have inverses.

Not all square matrices have inverses.

matrix A has an inverse A­1        |A| ≠ 0

singular matrices have no inverse

2x2 matrices:

or using GDC: x-1 key

3x3 matrices:

using GDC: x-1 key

Solutions of systems of linear equations

Using matrices we can rewrite these equations as:

A X = BA­1(AX) = A­1B(A­1 A) X  = A­1B

I   X  = A­1B

   X  = A­1B

   X  = A­1BA       X  =  B ⇒

X

⇒ x = 3  , y = 1 

Given the simultaneous equations:

write them in matrix form and find x, y and z.

using GDC :

Use your GDC to find the inverse of the matrix

Hence, solve the simultaneous equations:

4 11 5

1 4 2

1 2 1

4 x + 11 y + 5 z = 2    x + 4  y + 2 z =  1   x + 2  y + 1  z =  4

The matrix A =2 0 2

5 1 0

­1 4 a

a) Find an expression in terms of a for detA.

b) Find the value of a for which A-1 does not exist.

c) Solve the equation A when a = 0

giving your answers correct to 3 s.f.

Exercise Book page 336 Ex 1 a), 2 and 5 page 337 : EX8, 10,

revision exercise 12 page 339