Chapter 11.4, Problem 93AYU

To determine

To explain: The solution of the system of equations $AX=\varphi$ when $A^{-1}$ exist and when $A^{-1}$ does not exist.

Answer to Problem 93AYU

The solution of the system of equations $AX=\varphi$ when $A^{-1}$ exist is $X=A^{-1}$ and when $A^{-1}$ does not exist there is no solution.

Explanation of Solution

Given information:

The system of equations $AX=\varphi$ .

Consider system of equations $AX=\varphi$ .

Pre multiply both sides by $A^{-1}$ .

  $\begin{array}{c}{A}^{-1}AX=A^{-1}\varphi \\ \left(A^{-1}A\right)X=A^{-1}\varphi \end{array}$

For a matrix A, the inverse of a $2\times 2$ is given by the formula $A{A}^{-1}=I_2$ , where $I_2$ is the identity matrix.

  $\begin{array}{c}IX=A^{-1}\varphi \\ X=A^{-1}\varphi \\ X=A^{-1}\end{array}$

The above expression obtained is only valid if A is an invertible matrix. That is inverse of A exists. A is a nonsingular matrix. Determinant of A is not zero.

Thus, the solution of the system of equations $AX=\varphi$ when $A^{-1}$ exist is $X=A^{-1}$ and when $A^{-1}$ does not exist there is no solution.