Question

a c 0 1 2 1
b d * 2 -1 = -1 0 Find Matrix a, b, c and d

Answer 1

We have an equation:

$$\left( \begin{array}{cc} a & c \\ b & d \end{array} \right) \cdot \left( \begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array} \right) = \left( \begin{array}{cc} 2 & 1 \\ -1 & 0 \end{array} \right)$$

So solve this let's firstly find inverse matrix to $\left( \begin{array}{cc}0 & 1\\ 2 & -1 \end{array} \right)$.

$$\left( \begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array} \right)^{-1} = \left( \begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\ 1 & 0 \end{array} \right)$$

Now, let's multiply both parts of equation by inverse matrix:

$$\left( \begin{array}{cc} a & c \\ b & d \end{array} \right) \cdot \left( \begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array} \right)^{-1} = \left( \begin{array}{cc} 2 & 1 \\ -1 & 0 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \\ 2 & -1 \end{array} \right)^{-1}$$ $$\left( \begin{array}{cc} a & c \\ b & d \end{array} \right) = \left( \begin{array}{cc} 2 & 1 \\ -1 & 0 \end{array} \right) \left( \begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\ 1 & 0 \end{array} \right) = \left( \begin{array}{cc} 2 & 1 \\ -\frac{1}{2} & -\frac{1}{2} \end{array} \right)$$

So

$$\left( \begin{array}{cc} a & c \\ b & d \end{array} \right) = \left( \begin{array}{cc} 2 & 1 \\ -\frac{1}{2} & -\frac{1}{2} \end{array} \right)$$