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Problem 166

Find the inverse of A where $$\mathrm{A}=\mid \begin{array}{rr} 2 & 3 \mid \\ \mid 3 & 5 \mid \end{array}$$

Expert verified
The inverse of matrix A is: $$\mathrm{A}^{-1}=\mid \begin{array}{rr} 5 & -3 \mid \\\ \mid -3 & 2 \mid \end{array}$$
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Step 1: Find the determinant of matrix A

To find the determinant of a 2x2 matrix, we can use the following formula: $\det(A) = ad - bc$ For our given matrix A: $$\mathrm{A}=\mid \begin{array}{rr} 2 & 3 \mid \\\ \mid 3 & 5 \mid \end{array}$$ a = 2, b = 3, c = 3, and d = 5. So, the determinant of A is: $\det(A) = (2)(5) - (3)(3) = 10 - 9 = 1$

Step 2: Find the inverse of matrix A using the formula

Since the determinant is non-zero, matrix A has an inverse. The formula for finding the inverse of a 2x2 matrix is as follows: $A^{-1} = \frac{1}{\det(A)}\mid \begin{array}{cc} d & -b\\ -c & a \end{array}$ Using the values of a, b, c, d, and the determinant calculated in step 1: $A^{-1} = \frac{1}{1}\mid \begin{array}{cc} 5 & -3\\ -3 & 2 \end{array}$

Step 3: Simplify the inverse matrix

Since the determinant is equal to 1, the fraction in the formula doesn't affect the values of the matrix elements, and therefore the inverse of matrix A is: $$\mathrm{A}^{-1}=\mid \begin{array}{rr} 5 & -3 \mid \\\ \mid -3 & 2 \mid \end{array}$$

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