Solveeit Logo
Login

Question

Mathematics Question on Matrices

Find the inverse of each of the matrices,if it exists. [23 57]\begin{bmatrix} 2 & 3\\\ 5 & 7 \end{bmatrix}

Answer

Let A=[23 57]\begin{bmatrix} 2 & 3\\\ 5 & 7 \end{bmatrix}

We know that A=IAA = IA

⇒ [23 57]\begin{bmatrix} 2 & 3\\\ 5 & 7 \end{bmatrix} = [10 01]A\begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix}A

⇒ [132 57]\begin{bmatrix} 1 & \frac32\\\ 5 & 7 \end{bmatrix}=\begin{bmatrix} \frac12 & 0\\\ 0 & 1 \end{bmatrix}$$A (R1→(12R1))(R_1 \rightarrow (\frac{1}{2}R_1) )

⇒ [132 0−12]\begin{bmatrix} 1 & \frac32\\\ 0 & \frac{-1}{2} \end{bmatrix}= [120 −521]A\begin{bmatrix} \frac12 & 0\\\ -\frac{5}{2} & 1 \end{bmatrix}A (R2→R2−5R1)(R_2→ R_2-5R_1)

⇒ \begin{bmatrix} 1 & 0\\\ 0 & \frac{-1}{2} \end{bmatrix}$$=\begin{bmatrix} -7 & 3\\\ -\frac{5}{2} & 1 \end{bmatrix}A (R1→R1+3R2)(R_1→ R_1+3R_2)

⇒ [10 01]\begin{bmatrix} 1 & 0\\\ 0 & 1 \end{bmatrix} = [−73 5−2]A\begin{bmatrix} -7 & 3\\\ 5 & -2 \end{bmatrix}A (R2→−2R1)(R_2\rightarrow-2R_1)

A−1A^{-1}= [−73 5−2]\begin{bmatrix} -7 & 3\\\ 5 & -2 \end{bmatrix}