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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