Solve A^-1=1/2(A-E)

$$A ^ { - 1 } = \frac { 1 } { 2 } ( A - E )$$

Solve for E

$$E=A-\frac{2}{A}$$

$$A\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{2} by A-E.

$$A^{-1}=\frac{1}{2}A-\frac{1}{2}E$$

Swap sides so that all variable terms are on the left hand side.

$$\frac{1}{2}A-\frac{1}{2}E=A^{-1}$$

Subtract \frac{1}{2}A from both sides.

$$-\frac{1}{2}E=A^{-1}-\frac{1}{2}A$$

Reorder the terms.

$$-\frac{1}{2}E=-\frac{1}{2}A+\frac{1}{A}$$

Multiply both sides of the equation by 2A, the least common multiple of 2,A.

$$-\frac{1}{2}E\times 2A=-\frac{1}{2}A\times 2A+2$$

Multiply -\frac{1}{2} and 2 to get -1.

$$-EA=-\frac{1}{2}A\times 2A+2$$

Multiply A and A to get A^{2}.

$$-EA=-\frac{1}{2}A^{2}\times 2+2$$

Multiply -\frac{1}{2} and 2 to get -1.

$$-EA=-A^{2}+2$$

The equation is in standard form.

$$\left(-A\right)E=2-A^{2}$$

Divide both sides by -A.

$$\frac{\left(-A\right)E}{-A}=\frac{2-A^{2}}{-A}$$

Dividing by -A undoes the multiplication by -A.

$$E=\frac{2-A^{2}}{-A}$$

Divide -A^{2}+2 by -A.

$$E=A-\frac{2}{A}$$

Solve for A

$$A=\frac{-\sqrt{E^{2}+8}+E}{2}$$

$$A=\frac{\sqrt{E^{2}+8}+E}{2}$$