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The inverse of a2×2matrix

sigma-matrices7-2009-1 Once you know how to multiply matrices it is natural to ask whether they can be divided. The answer is no. However, by defining another matrix called theinverse matrixit is possible to work

with an operation which plays a similar role to division. In this leaflet we explain what is meant by

an inverse matrix and how the inverse of a2×2matrix is calculated.

Preliminary example

Suppose we calculate the product of the two matrices?4 31 1? and?1-3 -1 4? 4 3

1 1? ?

1-3 -1 4? =?1 00 1? If we re-order the matrices and recalculate we will obtain the same result. You should verify this: ?1-3 -1 4? ? 4 3 1 1? =?1 00 1? Note that the result of multiplying the two matrices together is theidentitymatrix. Pairs of square matrices which have this property are calledinversematrices. The first is the inverse of the second, and vice-versa.

The inverse of a2×2matrix

Theinverseof a2×2matrixA, is another2×2matrix denoted byA-1with the property that AA -1=A-1A=I whereIis the2×2identity matrix?1 00 1? . That is, multiplying a matrix by its inverse produces an identity matrix. Note that in this contextA-1does not mean1 A. Not all2×2matrices have an inverse matrix. If the determinant of the matrix is zero, then it will

not have an inverse; the matrix is then said to besingular. Only non-singular matrices have inverses.

A simple formula for the inverse

In the case of a2×2matrixA=?a b

c d? a simple formula exists to find its inverse: ifA=?a b c d? thenA-1=1ad-bc? d-b -c a? Note that the quantityad-bcis the determinant ofA. Furthermore,1ad-bcis not defined when ad-bc= 0since it is never possible to divide by zero. It is for this reason that the inverse ofA does not exist if the determinant ofAis zero. www.mathcentre.ac.uk 1 c?mathcentre 2009

ExampleFind the inverse of the matrixA=?3 14 2?

Solution

Using the formula

A -1=1 (3)(2)-(1)(4)? 2-1 -4 3? 1 2? 2-1 -4 3?

This could be written as

?1-1 2 -23 2? You should check that this answer is correct by performing the matrix multiplicationAA-1. The result should be the identity matrixI=?1 00 1?

Example

Find the inverse of the matrixA=?2 4

-3 1?

Solution

Using the formula

A -1=1 (2)(1)-(4)(-3)? 1-4 3 2? 1 14? 1-4 3 2?

This can be written

A -1=?1/14-4/14

3/14 2/14?

=?1/14-2/7

3/14 1/7?

although it is quite permissible to leave the factor 1

14at the front of the matrix.

Example

Find, if possible, the inverse of the matrixA=?3 26 4?

Solution

In this case the determinant of the matrix is zero: ?3 26 4????? = 3×4-2×6 = 0 Because the determinant is zero the matrix is singular and noinverse exists. We explain how to find the inverse of a3×3matrix in a later leaflet in this series. Note that a video tutorial covering the content of this leaflet is available fromsigma. www.mathcentre.ac.uk 2 c?mathcentre 2009quotesdbs_dbs14.pdfusesText_20