Let's analyze and solve this step by step.

The given problem starts with:

• $\text{det} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = -3$.

We are tasked to find: $\text{det} \begin{bmatrix} a + c & b + d \\ c - 2a & d - 2b \end{bmatrix}.$

Step 1: Recall the determinant formula

The determinant of a $2 \times 2$ matrix $\begin{bmatrix} p & q \\ r & s \end{bmatrix}$ is given by: $\text{det} = ps - qr.$

So for the matrix $\begin{bmatrix} a + c & b + d \\ c - 2a & d - 2b \end{bmatrix}$, the determinant is: $\text{det} = (a + c)(d - 2b) - (b + d)(c - 2a).$

Step 2: Expand the determinant

1. Expand the first term $(a + c)(d - 2b)$: $(a + c)(d - 2b) = ad - 2ab + cd - 2cb.$

2. Expand the second term $(b + d)(c - 2a)$: $(b + d)(c - 2a) = bc - 2ab + dc - 2ad.$

Thus, the determinant becomes: $\text{det} = (ad - 2ab + cd - 2cb) - (bc - 2ab + dc - 2ad).$

Step 3: Combine like terms

Group the terms systematically:

1. Coefficients of $ad$: $ad - (-2ad) = 3ad$.
2. Coefficients of $ab$: $-2ab - (-2ab) = 0$.
3. Coefficients of $cd$: $cd - dc = 0$.
4. Coefficients of $bc$: $-2cb - bc = -3bc$.

The determinant simplifies to: $\text{det} = 3ad - 3bc.$

Step 4: Substitute the original determinant value

From the original matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, we know: $\text{det} = ad - bc = -3.$

Substitute $ad - bc = -3$ into the simplified determinant expression: $\text{det} = 3(ad - bc) = 3(-3) = -9.$

Final Answer:

$\text{det} \begin{bmatrix} a + c & b + d \\ c - 2a & d - 2b \end{bmatrix} = -9.$

Would you like a more detailed explanation, or do you have any questions?

Related Questions:

1. How do you compute determinants for larger matrices?
2. What are properties of determinants that simplify calculations?
3. How does the determinant change when rows or columns are scaled or swapped?
4. Can you explain the geometric interpretation of a determinant?
5. What is the relationship between the determinant and matrix invertibility?

Tip:

When simplifying expressions with determinants, always look for opportunities to factor or group terms related to the original determinant for efficiency.