MATRICES : MATRICES
NOTATION OF A MATRIX : NOTATION OF A MATRIX A matrix is written with ( ) or [ ] brackets.
Do not confuse a matrix with a determinant which uses vertical bars | |.
A matrix is a pattern of numbers; a determinant gives us a single number.
The size of a matrix is written: rows × columns.
2 4 6
7 6 5
It has 2 rows and 3 columns.
Application of Matrices : Application of Matrices Matrices are used to solve problems in:
1. simultaneous equations
2. electronics
3. statics
4. robotics
5. linear programming
6. optimisation
7. intersections of planes
8. genetics
ELEMENTS IN A MATRIX : ELEMENTS IN A MATRIX The elements in a matrix A are denoted by aij, where i is the row number and j is the column number.
Example:
Consider 4 5 7
6 8 4
The element a21 = 6, since the element in the 2nd row and 1st column is 6.
The element a13 = 7, since the element in the 1st row and 3rd column is 7.
EQUALITY OF MATRICES : EQUALITY OF MATRICES Equal matrices have identical corresponding elements.
Example:
If 3 x 3 5
y 2 = 9 2
then x = 5, y = 9.
ADDITION & SUBTRACTION OF MATRICES : ADDITION & SUBTRACTION OF MATRICES We can only add (or subtract) matrices if they have the same dimensions. That is, the two matrices must have the same number of rows and the same number of columns.
To add matrices, just add corresponding elements:
IDENTITY MATRIX : IDENTITY MATRIX A matrix with equal numbers of rows and columns is called Square matrix.
The Identity Matrix, written I, is a square matrix where all the elements are 0 except the principal diagonal which has all ones.
The identity matrix is also known as the
unit matrix.
Eg: 2x2 matrix 1 0 0 3 x 3 matrix
1 0 0 1 0
0 1 0 0 1
DIAGONAL MATRIX : DIAGONAL MATRIX A diagonal matrix is a square matrix that has zeroes everywhere except along the main diagonal (top left to bottom right).
Example:
4 0 0
0 7 0
0 0 6
The identity matrix is also an example .
SCALAR MULTIPLICATION : SCALAR MULTIPLICATION We multiply (or divide) each element by the scalar value (a single number).
Example:
A = 1 5
2 6
2A = 2 1 5 = 2 10
2 6 4 12
MULTIPLICATION OF MATRICES : MULTIPLICATION OF MATRICES We can only multiply matrices if the number of columns in the first matrix is the same as the number of rows in the second matrix.
EXAMPLE:
Multiplying a 2 × 3 matrix by a 3 × 2 matrix is possible and it gives a 2 × 2 matrix as the answer.
Method to Multiply 2 matrices : Method to Multiply 2 matrices We multiply and add the elements as follows. We work across the 1st row of the first matrix, multiplying down the 1st column of the second matrix, element by element. We add the resulting products. Our answer goes in position a11 (top left) of the answer matrix.
We do a similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result is placed in position a12.
Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result is placed in position a21.
Finally, we do the 2nd row of the first matrix and the 2nd column of the second matrix. The result is placed in position a22.