Question

A matrix \(A\) and a vector \(\vec{x}\) are given. Find the product \(A \vec{x}\). $$ A=\left[\begin{array}{ccc} -2 & 0 & 3 \\ 1 & 1 & -2 \\ 4 & 2 & -1 \end{array}\right], \quad \vec{x}=\left[\begin{array}{l} 4 \\ 3 \\ 1 \end{array}\right] $$

Short answer

The product \( A\vec{x} \) is \( \begin{bmatrix} -5 \\ 5 \\ 21 \end{bmatrix} \).

Step-by-step solution

Matrix and Vector Format

First, understand the format of the given matrix \( A \) and vector \( \vec{x} \). Matrix \( A \) is a 3x3 matrix, which is represented as \( \begin{bmatrix} -2 & 0 & 3 \ 1 & 1 & -2 \ 4 & 2 & -1 \end{bmatrix} \). The vector \( \vec{x} \) is a column vector represented as \( \begin{bmatrix} 4 \ 3 \ 1 \end{bmatrix} \).

Compute the First Element of the Product

To find the first element of the resulting vector from the product \( A\vec{x} \), perform the following calculation: multiply the first row of matrix \( A \) by vector \( \vec{x} \) and then sum the products. \(-2 \cdot 4 + 0 \cdot 3 + 3 \cdot 1 = -8 + 0 + 3 = -5\).

Compute the Second Element of the Product

For the second element, calculate as follows: multiply the second row of matrix \( A \) by vector \( \vec{x} \) and sum the results. \( 1 \cdot 4 + 1 \cdot 3 + (-2) \cdot 1 = 4 + 3 - 2 = 5 \).

Compute the Third Element of the Product

For the third element, apply the same process: multiply the third row of matrix \( A \) by vector \( \vec{x} \) and sum the products. \( 4 \cdot 4 + 2 \cdot 3 + (-1) \cdot 1 = 16 + 6 - 1 = 21 \).

Assemble the Resulting Vector

Finally, combine the results from steps 2, 3, and 4 to form the resulting vector from the matrix-vector multiplication: \( A\vec{x} = \begin{bmatrix} -5 \ 5 \ 21 \end{bmatrix} \).

Key concepts

Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, matrices, and linear transformations. It is a key area of study for various fields such as physics, computer science, and engineering. The essence of linear algebra revolves around solving systems of linear equations and understanding vector spaces and matrix theory.

Matrices and vectors form the foundation of linear algebra. A matrix is an array of numbers arranged in rows and columns, while a vector is a single column (or row) of numbers. Linear algebra uses these to represent linear transformations and perform operations like matrix-vector multiplication.

Linear algebraic tools allow for the modeling and solving of complex problems, helping us to find solutions efficiently. In practical applications, these concepts help in performing tasks like image processing, computational simulations, and data analysis.

Matrix Operations

Matrix operations are mathematical procedures performed on matrices, such as addition, subtraction, and multiplication. They enable us to manipulate matrices to achieve a desired outcome, such as solving systems of equations or transforming geometric data.

One of the most important matrix operations is matrix-vector multiplication. In this operation, each element of a vector is multiplied by the corresponding elements in a row of the matrix, and the results are summed to produce a new vector. The resulting vector has as many elements as there are rows in the matrix. This operation is fundamental for transformations and solving linear equations.

Other matrix operations include:

• Transpose: Rearranging the elements of a matrix by flipping it over its diagonal.
• Inverse: A matrix that, when multiplied by the original matrix, yields the identity matrix.
• Determinant: A scalar value that provides information about the matrix, including whether an inverse exists.

Understanding these operations is crucial for mastering linear algebra.

Vector Arithmetic

Vector arithmetic involves operations such as addition, subtraction, and scalar multiplication between vectors. Vectors are simply quantities that have both magnitude and direction, which makes them useful in physics and engineering to represent forces and velocities.

In vector arithmetic:

• Addition: Vectors are added by summing their respective components. For example, if \(\vec{a} = \begin{bmatrix} a_1 \ a_2 \ a_3 \end{bmatrix}\\) and \(\vec{b} = \begin{bmatrix} b_1 \ b_2 \ b_3 \end{bmatrix}\\), the sum is \(\vec{a} + \vec{b} = \begin{bmatrix} a_1 + b_1 \ a_2 + b_2 \ a_3 + b_3 \end{bmatrix}\\).
• Subtraction: Similar to addition, with subtraction applied to each component.
• Scalar Multiplication: Each element of the vector is multiplied by a scalar (a single number).

When performing operations like matrix-vector multiplication, understanding vector arithmetic is essential as it facilitates combining elements and understanding the geometric implications of these operations.