Question 4.1.4: Consider the 2 × 2 matrix A defined by A = [ a b c d ] and c......

The inverse of a square matrix A is a matrix of the same dimension, denoted by A^{-1}, such that

A A^{-1}=A^{-1} A=I

where I is the identity matrix. In this case I has the form

I=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right]

The inverse matrix for a general 2 \times 2 matrix is

A^{-1}=\frac{1}{\operatorname{det} A}\left[\begin{array}{rr} d & -b \\ -c & a \end{array}\right]          (4.18)

provided that \operatorname{det} A \neq 0, where \operatorname{det} A denotes the determinant of the matrix A. The determinant of the matrix A has the value

\operatorname{det} A=a d-b c

To see that equation (4.18) is in fact the inverse, note that

\begin{aligned} A^{-1} A & =\frac{1}{a d-b c}\left[\begin{array}{rr} d & -b \\ -c & a \end{array}\right]\left[\begin{array}{ll} a & b \\ c & d \end{array}\right] \\ & =\frac{1}{a d-b c}\left[\begin{array}{rr} a d-b c & b d-b d \\ a c-a c & a d-b c \end{array}\right]=\left[\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right] \end{aligned}

It is important to realize that the matrix A has an inverse if and only if \operatorname{det} A \neq 0. Thus, requiring det A=0 forces A not to have an inverse. Matrices that do not have an inverse are called singular matrices. Note that if the matrix A is symmetric, c=b and A^{-1} is also symmetric.