Inverse of 2x2 matrix:
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 the inverse matrix it 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 a 2 × 2 matrix is calculated.
The inverse of a 2 × 2 matrix A, is another 2 × 2 matrix denoted by A −1 with the property that
AA−1 = A −1A = I
where I is the 2 × 2 identity matrix\[\begin{bmatrix} 1&0 \\ 0 & 1\\ \end{bmatrix}\]. That is, multiplying a matrix by its inverse produces an identity matrix. Note that in this context A−1 does not mean 1 A . Not all 2 × 2 matrices 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 be singular. Only non-singular matrices have inverses.
simple formula for inverse:
In the case of a 2 × 2 matrix A =\[\begin{bmatrix} a& b \\ c & d\\ \end{bmatrix}\] a simple formula exists to find its inverse:
if A=\[\begin{bmatrix} a& b \\ c & d\\ \end{bmatrix}\] then A −1 = \[ \frac{1 }{ ad-bc} \] \[\begin{bmatrix} d& -b \\ -c & a\\ \end{bmatrix}\]
Note that the quantity ad − bc is the determinant of A. Furthermore, 1 / ad−bc is not defined when ad − bc = 0 since it is never possible to divide by zero. It is for this reason that the inverse of A does not exist if the determinant of A is zero.
Calculating the Inverse 3x3 Matrix by the following steps:
Step 1: calculating the Matrix of Minors,
Step 2: then turn that into the Matrix of Cofactors,
Step 3: then the Adjugate, and
Step 4: multiply that by 1/Determinant.
Example: Find the inverse of A:
A=\[\begin{bmatrix} 3 & 0 &2 \\ 2 & 0 & -2 \\ 0 & 1 &1 \\ \end{bmatrix}\]
Step 1: Matrix of Minors
The first step is to create a "Matrix of Minors". This step has the most calculations.
For each element of the matrix:
ignore the values on the current row and column
calculate the determinant of the remaining values
Put those determinants into a matrix (the "Matrix of Minors")
=\[\begin{bmatrix} 0*1-(-2)*1& 2*1-(-2)*0 &2 *1-0*0 \\ 0*1-2*1& 3*1-2*0& 3 *1-0*0\\ 0*(-2)-2*0 & 3*(-2)- 2*0&3*0-0*2 \\ \end{bmatrix}=\begin{bmatrix} 3 & 0 &2 \\ 2 & 0 & -2 \\ 0 & 1 &1 \\ \end{bmatrix}\]
and this is the minor of matrix.
Step 2: Matrix of Cofactors
We need to change the sign of alternate cells, and apply this to minor of matrix.
minor of matrix \[\begin{bmatrix} 2& 2 & 2 \\ -2 &3 &3 \\ 0& -10 &0 \\ \end{bmatrix}\rightarrow \begin{bmatrix} + & -& + \\ - & + & - \\ + & - &+ \\\end{bmatrix}\rightarrow \begin{bmatrix}2 &-2 &2 \\ 2 &3 &-10 \\0 &3 &0 \\ \end{bmatrix}\] matrix cofactors
Step 3: Adjugate (also called Adjoint)
Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same):
\[\begin{bmatrix} 2& 2& 0 \\ -2& 3 & 10 \\ 2& -3 &0 \\ \end{bmatrix}\]
Step 4: Multiply by 1/Determinant
Now find the determinant of the original matrix.
\[A^{-1}=1/10\begin{bmatrix} 2& 2& 0 \\ -2& 3 & 10 \\ 2& -3 &0 \\ \end{bmatrix}=\begin{bmatrix} 0. 2& 0.2& 0 \\ -0.2& 0.3 & 1 \\ 0. 2& -0.3 &0 \\ \end{bmatrix}\] .
It is exactly the same steps for larger matrices (such as a 4×4, 5×5, etc).
