Find the sum of the entries in the inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$

1. Analyze the Question

We are given a $2 \times 2$ matrix $A = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}$. We need to find its inverse, $A^{-1}$, and then calculate the sum of all the elements (entries) within that resulting matrix.

2. Explain the Concept

For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the inverse is given by the formula:

Where $\det(A) = (ad - bc)$.

3. Show the Solution

Step 1: Find the determinant of A.

Step 2: Find the inverse matrix. Using the formula:

Multiply each element by $-1$:

Step 3: Find the sum of the entries. Sum $= (-5) + 2 + 3 + (-1)$ Sum $= -5 + 2 + 3 - 1$ Sum $= -3 + 3 - 1$ Sum $= 0 - 1 = -1$

4. Identify the Answer

The sum of the entries is $-1$. Comparing this to the given options, it matches option B.

5. Exam Tip

When finding the inverse of a $2 \times 2$ matrix, remember the "Swap and Negate" rule: swap the elements on the leading diagonal ($a$ and $d$) and change the signs of the elements on the other diagonal ($b$ and $c$). Always double-check your determinant calculation, as a sign error there will change the entire result!