Inverse Matrices (2 x 2) How to find the inverse of a 2x2 matrix

Inverse of a number Solving the equation 3x = 5 If x is a natural number then the process is: Divide both sides by 3 X = 5 / 3 Alternatively multiply both sides by: 1/31/3

Inverse of a Matrix Solving the equation Ax = B If A is a matrix then the process is not so simple: How do you divide by a matrix

Inverse of a number When we are talking about our natural numbers, the inverse of a number is it’s reciprocal. When we multiply a number by it’s inverse we get 1. For example:

Inverse of a matrix What do you think we would get if we multiplied a matrix by it’s inverse? A matrix multiplied by its inverse always gives us an identity matrix.

Let A -1 = Multiplying out gives.. So AA -1 = I So A –1 = Solvingc = –  d =  3c + d = 0 2c + 2d = 1 3a + b = 1 2a + 2b = 0 Solvinga =  b = –  Find A -1 Rows × Columns What do you notice

Finding Inverses 2x2 In words: Take the original matrix. Switch a and d. Change the signs of b and c. Multiply the new matrix by 1 over the determinant of the original matrix. a  d – b  c is called the determinant of A. Det A = |A|= ad - bc Link to notes

= Example: Find the inverse of A. Finding Inverses 2x2

Inverse of a matrix Not all matrices have an inverse. If the determinant of a matrix is 0, then it has no inverse and is said to be SINGULAR. All others are said to be NON-SINGULAR

Finding Inverses 2x2 Link to notes

Effect on Area of Determinant The matrixis applied to the unit square. Determine the area of the transformed square.

The area of the parallelogram is sq units. The determinant has transformed the area by a factor of DetA = |A| = 5  2 – 2  3 = 4 4 Link to notes Link to Spreadsheet x x x x 4