Solve aAx=b+lambdaa

$$a A x = b + \lambda a$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=\frac{a\lambda +b}{ax}\text{, }&x\neq 0\text{ and }a\neq 0\\A\in \mathrm{C}\text{, }&\left(b=-a\lambda \text{ and }x=0\right)\text{ or }\left(b=0\text{ and }a=0\text{ and }x\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$axA=a\lambda +b$$

Divide both sides by ax.

$$\frac{axA}{ax}=\frac{a\lambda +b}{ax}$$

Dividing by ax undoes the multiplication by ax.

$$A=\frac{a\lambda +b}{ax}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{b}{Ax-\lambda }\text{, }&\lambda \neq Ax\\a\in \mathrm{C}\text{, }&b=0\text{ and }\lambda =Ax\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \lambda a from both sides.

$$aAx-\lambda a=b$$

Combine all terms containing a.

$$\left(Ax-\lambda \right)a=b$$

Divide both sides by Ax-\lambda .

$$\frac{\left(Ax-\lambda \right)a}{Ax-\lambda }=\frac{b}{Ax-\lambda }$$

Dividing by Ax-\lambda undoes the multiplication by Ax-\lambda .

$$a=\frac{b}{Ax-\lambda }$$

Solve for A

$$\left\{\begin{matrix}A=\frac{a\lambda +b}{ax}\text{, }&x\neq 0\text{ and }a\neq 0\\A\in \mathrm{R}\text{, }&\left(b=-a\lambda \text{ and }x=0\right)\text{ or }\left(b=0\text{ and }a=0\text{ and }x\neq 0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

The equation is in standard form.

$$axA=a\lambda +b$$

Divide both sides by ax.

$$\frac{axA}{ax}=\frac{a\lambda +b}{ax}$$

Dividing by ax undoes the multiplication by ax.

$$A=\frac{a\lambda +b}{ax}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{b}{Ax-\lambda }\text{, }&\lambda \neq Ax\\a\in \mathrm{R}\text{, }&b=0\text{ and }\lambda =Ax\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract \lambda a from both sides.

$$aAx-\lambda a=b$$

Combine all terms containing a.

$$\left(Ax-\lambda \right)a=b$$

Divide both sides by Ax-\lambda .

$$\frac{\left(Ax-\lambda \right)a}{Ax-\lambda }=\frac{b}{Ax-\lambda }$$

Dividing by Ax-\lambda undoes the multiplication by Ax-\lambda .

$$a=\frac{b}{Ax-\lambda }$$