QuestionIf $A$ and $B$ are matrices and $A B=I$, which of the following represents the value of $B$ ? $$A=\left[\begin{array}{cc} -1 & 4 \\ -3 & 8 \end{array}\right]$$ $\left[\begin{array}{cc}\frac{1}{4} & 1 \\ -\frac{3}{4} & -2\end{array}\right]$ $\left[\begin{array}{cc}-\frac{1}{4} & -1 \\ \frac{3}{4} & 2\end{array}\right]$ $\left[\begin{array}{cc}-2 & 1 \\ -\frac{3}{4} & \frac{1}{4}\end{array}\right]$ $\left[\begin{array}{rr}2 & -1 \\ \frac{3}{4} & -\frac{1}{4}\end{array}\right]$

Studdy Solution
Compare the calculated inverse $A^{-1}$ with the given options:
Option 1: $$\begin{bmatrix} \frac{1}{4} & 1 \\ -\frac{3}{4} & -2 \end{bmatrix}$$
Option 2: $$\begin{bmatrix} -\frac{1}{4} & -1 \\ \frac{3}{4} & 2 \end{bmatrix}$$
Option 3: $$\begin{bmatrix} -2 & 1 \\ -\frac{3}{4} & \frac{1}{4} \end{bmatrix}$$
Option 4: $$\begin{bmatrix} 2 & -1 \\ \frac{3}{4} & -\frac{1}{4} \end{bmatrix}$$
The correct option is Option 4, which matches $A^{-1}$.
The value of $B$ is:
$$\boxed{\begin{bmatrix} 2 & -1 \\ \frac{3}{4} & -\frac{1}{4} \end{bmatrix}}$$