Question

Identify the inverse of the given matrix, if it exists.

Answer 1

To find the inverse matrix, augment it with the identity matrix and perform row operations trying to make the identity matrix to the left. Then to the right will be the inverse matrix.

Augment the given matrix with the identity matrix

`\mleft[\begin{array}cc3 & -4 & 1 & 0 \\ 5 & -8 & 0 & 1\end{array}\mright]`
`\begin{gathered} R_1=(R_(1))/(3)\mleft[\begin{array}cc1 & -(4)/(3) & (1)/(3) & 0 \\ 5 & -8 & 0 & 1\end{array}\mright] \\ R_2=R_2-5R_1\mleft[\begin{array}cc1 & -(4)/(3) & (1)/(3) & 0 \\ 0 & -(4)/(3) & -(5)/(3) & 1\end{array}\mright] \\ R_2=-(3 R_(2))/(4)\mleft[\begin{array}cc1 & -(4)/(3) & (1)/(3) & 0 \\ 0 & 1 & (5)/(4) & -(3)/(4)\end{array}\mright] \\ R_1=R_1+(4 R_(2))/(3)\mleft[\begin{array}cc1 & 0 & 2 & -1 \\ 0 & 1 & (5)/(4) & -(3)/(4)\end{array}\mright] \end{gathered}`

Finally, on the right is the inverse matrix. Therefore the inverse matrix is:

`\mleft[\begin{array}{cc}2 & -1 \\ (5)/(4) & -(3)/(4)\end{array}\mright]`