Compute the product Av where
A=[−254−1−31]v=⎡⎣⎢−301⎤⎦⎥

We are given the matrix \( A \) and the vector \( v \), and we need to compute the product \( Av \).

Matrix \( A \) and vector \( v \) are as follows:

\[
A = \begin{bmatrix}
-2 & 5 & 4 \\
-1 & -3 & 1
\end{bmatrix}, \quad
v = \begin{bmatrix}
-3 \\
0 \\
1
\end{bmatrix}
\]

To compute the product \( Av \), we multiply matrix \( A \) by vector \( v \):

\[
Av = \begin{bmatrix}
-2 & 5 & 4 \\
-1 & -3 & 1
\end{bmatrix} 
\begin{bmatrix}
-3 \\
0 \\
1
\end{bmatrix}
\]

Let's compute each element of the resulting vector.

### First element of \( Av \):
\[
(-2)(-3) + (5)(0) + (4)(1) = 6 + 0 + 4 = 10
\]

### Second element of \( Av \):
\[
(-1)(-3) + (-3)(0) + (1)(1) = 3 + 0 + 1 = 4
\]

Thus, the product \( Av \) is:

\[
Av = \begin{bmatrix}
10 \\
4
\end{bmatrix}
\]

### Final Answer:
\[
Av = \begin{bmatrix}
10 \\
4
\end{bmatrix}
\]

Let me know if you'd like a detailed explanation of any step.

Here are five related questions for further exploration:
1. What is the dot product of two vectors?
2. How do you find the determinant of a 2x2 matrix?
3. Can you multiply a 3x3 matrix by a vector? How does it differ?
4. What are the properties of matrix-vector multiplication?
5. How does the rank of a matrix affect the solutions to the equation \(Ax = b\)?

**Tip**: When multiplying a matrix by a vector, ensure the number of columns in the matrix matches the number of elements in the vector to perform valid multiplication.

Compute the product Av where A=[−2 5 4; −1 −3 1] and v=[−3; 0; 1]

Learn how to compute the matrix-vector product Av where A=[-2 5 4; -1 -3 1] and v=[-3 0 1]. This solution involves key linear algebra concepts and shows step-by-step multiplication. Ideal for high school students in Grades 10-12.

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