Solve A=A^-1

$$A = A ^ { - 1 }$$

Solve for A

$$A=1$$

$$A=-1$$

Steps Using Difference of Squares

Subtract A^{-1} from both sides.

$$A-A^{-1}=0$$

Reorder the terms.

$$A-\frac{1}{A}=0$$

Variable A cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by A.

$$AA-1=0$$

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

$$A^{2}-1=0$$

Consider A^{2}-1. Rewrite A^{2}-1 as A^{2}-1^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(A-1\right)\left(A+1\right)=0$$

To find equation solutions, solve A-1=0 and A+1=0.

$$A=1 A=-1$$

Steps by Finding Square Root

Subtract A^{-1} from both sides.

$$A-A^{-1}=0$$

Reorder the terms.

$$A-\frac{1}{A}=0$$

Variable A cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by A.

$$AA-1=0$$

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

$$A^{2}-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$A^{2}=1$$

Take the square root of both sides of the equation.

$$A=1 A=-1$$

Steps Using the Quadratic Formula

Subtract A^{-1} from both sides.

$$A-A^{-1}=0$$

Reorder the terms.

$$A-\frac{1}{A}=0$$

Variable A cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by A.

$$AA-1=0$$

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

$$A^{2}-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 0 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$A=\frac{0±\sqrt{0^{2}-4\left(-1\right)}}{2}$$

Square 0.

$$A=\frac{0±\sqrt{-4\left(-1\right)}}{2}$$

Multiply -4 times -1.

$$A=\frac{0±\sqrt{4}}{2}$$

Take the square root of 4.

$$A=\frac{0±2}{2}$$

Now solve the equation A=\frac{0±2}{2} when ± is plus. Divide 2 by 2.

$$A=1$$

Now solve the equation A=\frac{0±2}{2} when ± is minus. Divide -2 by 2.

$$A=-1$$

The equation is now solved.

$$A=1 A=-1$$