Question

Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}\)

Answer 1

Let A=\(\begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}\)

We know that \(A = IA\) 

\(\begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}\)\(=A\)\(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

⇒ \(\begin{bmatrix} 1 & -1 \\ 0 & 2 \end{bmatrix}\)\(= A\)\(\begin{bmatrix} 1 & 0 \\ 2 & 1 \end{bmatrix}\)        \((C_1\rightarrow C_1+2C_2)\) 

⇒ \(\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}\)\(=A\)\(\begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}\)          \((C_2\rightarrow C_2+C_1)\) 

⇒ \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)\(=A\) \(\begin{bmatrix} 1 & \frac12 \\ 2 & \frac32 \end{bmatrix}\)          \((C_2\rightarrow \frac{1}{2}C_2) \)

\(\therefore A^{-1}=\) \(\begin{bmatrix} 1 & \frac12 \\ 2 & \frac32 \end{bmatrix}\)