Matrices form a mathematical system with operations such as multiplication and addition.


In general, the multiplication of 2 X 2 matrices is NOT commutative. i.e. The order in which the matrices are multiplied is important.


If A = Y10_Matrix_Properties_01.gif and B = Y10_Matrix_Properties_02.gif then

AB = Y10_Matrix_Properties_03.gifY10_Matrix_Properties_04.gif = Y10_Matrix_Properties_05.gif

BA = Y10_Matrix_Properties_06.gifY10_Matrix_Properties_07.gif=Y10_Matrix_Properties_08.gif

Showing that AB ≠ BA.

The same applies to matrix subtraction A − B ≠ B − A.

Matrix subtraction is NOT commutative.

Matrix addition is commutative A + B = B + A.


Matrix multiplication is associative. i.e. The grouping of the matrices is not important when multiplying.


If P =Y10_Matrix_Properties_09.gif , Q = Y10_Matrix_Properties_10.gifR = Y10_Matrix_Properties_11.gif



Showing the P(QR) = (PQ)R

Identity Matrix

The identity 2 X 2 matrix for addition is Y10_Matrix_Properties_14.gif.

e.g. Y10_Matrix_Properties_15.gif

The identity 2 X 2 matrix for multiplication is Y10_Matrix_Properties_16.gif

e.g. Y10_Matrix_Properties_17.gif

Inverse Matrix

Addition The inverse 2 X 2 matrix for addition is obtained by changing the signs of each element.

e.g. If A = Y10_Matrix_Properties_18.gif, the additive inverse is Y10_Matrix_Properties_19.gif

A matrix and its inverse combine to give the identity matrix.

e.g. Y10_Matrix_Properties_20.gif

Multiplication The inverse 2 X 2 matrix for multiplication is obtained by:

  • Exchanging the elements of the leading diagonal.
  • Changing the signs of the elements of the other diagonal.
  • Multiplying the resulting matrix by Y10_Matrix_Properties_21.gif

In general terms, if A = Y10_Matrix_Properties_22.gif

the inverse A-1 = Y10_Matrix_Properties_23.gif


If B = Y10_Matrix_Properties_24.gif

The inverse of B, B-1 Y10_Matrix_Properties_25.gif

B-1 Y10_Matrix_Properties_26.gif

A matrix and its inverse combine to give the identity matrix.

e.g. B x B-1 = I


Solving Simultaneous Equations using Matrices

In addition to the four methods mentioned in an earlier topic, simultaneous equations can also be solved using matrices. The method involves writing the equations in matrix form and then multiplying each side by the inverse matrix to obtain the identity matrix.

The example below is used to solve the simultaneous equations:

2x + 3y = 5
x + 2y = 1

Method Example
Step 1: Set up the equations in matrix form. Y10_Matrix_Properties_28.gif
Step 2: Pre-multiply each side by the inverse 2 X 2 matrix. Y10_Matrix_Properties_29.gif

Step 3: Evaluate the matrix products.


The solution set is {(7, -3)}