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Problem 156

Find the inverse of $$\begin{array}{ll} \mid 1 & 2 \\ \mid 3 & 7 \end{array} \mid$$

Expert verified
The inverse of the given matrix is $$A^{-1} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}$$.
See the step by step solution

Step 1: Find the determinant

Let the matrix be $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, in this case, $$a=1, b=2, c=3$$, and $$d=7$$. The determinant of A is calculated as: $Det(A) = ad - bc$ $Det(A) = (1)(7) - (2)(3) = 7 - 6 = 1$

Step 2: Switch the elements at the main diagonal

Switch the a and d elements in the matrix A: $A' = \begin{bmatrix} 7 & 2 \\ 3 & 1 \end{bmatrix}$

Step 3: Change the sign of the elements at the other diagonal

Change the sign of the b and c elements in the matrix A': $A'' = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}$

Step 4: Divide the matrix by the determinant

Since the determinant is 1, we divide the matrix A'' by the determinant: $A^{-1} = \frac{1}{1}A'' = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}$ The inverse of the given matrix is: $A^{-1} = \begin{bmatrix} 7 & -2 \\ -3 & 1 \end{bmatrix}$

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