Question

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

Short answer

The product \(A \vec{x}\) is \(\begin{bmatrix} 8 \\ 7 \\ 3 \end{bmatrix}\).

Step-by-step solution

Understand the Problem

We need to find the matrix-vector product \(A \vec{x}\), where \(A\) is a \(3 \times 3\) matrix and \(\vec{x}\) is a \(3 \times 1\) vector.

Setup the Multiplication

To find the product \(A \vec{x}\), we will multiply each row of matrix \(A\) by the vector \(\vec{x}\) and sum up the results.

Multiply the First Row by the Vector

Calculate the first component of the result: \(a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3} = 2 \times 1 + 0 \times 4 + 3 \times 2 = 2 + 0 + 6 = 8\).

Multiply the Second Row by the Vector

Calculate the second component of the result: \(a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} = 1 \times 1 + 1 \times 4 + 1 \times 2 = 1 + 4 + 2 = 7\).

Multiply the Third Row by the Vector

Calculate the third component of the result: \(a_{31}x_{1} + a_{32}x_{2} + a_{33}x_{3} = 3 \times 1 + (-1) \times 4 + 2 \times 2 = 3 - 4 + 4 = 3\).

Write the Resulting Vector

Combine the results from each row to form the product vector: \[A \vec{x} = \begin{bmatrix} 8 \ 7 \ 3 \end{bmatrix}\].

Key concepts

Matrix Algebra

Matrix algebra is a significant part of mathematics that deals with matrices, which are rectangular arrays of numbers, symbols, or expressions. Understanding how matrices work is crucial for various applications, especially in solving linear equations and transformations in physics and engineering. Matrices can be added, subtracted, multiplied, and more, just like numbers, but they come with their own unique set of rules.

Consider the matrix \[A=\begin{bmatrix}2 & 0 & 3 \1 & 1 & 1 \3 & -1 & 2\end{bmatrix}. \]This is a 3x3 matrix, indicating it has 3 rows and 3 columns. The arrangement of numbers in rows and columns gives each element a unique position. Each element can be referred to using two indices, for example, the element in the first row and second column is 0, labeled as \(a_{12}\).

In matrix algebra, multiplication is not as straightforward as with regular numbers. For a matrix \(A\) to be multiplied by another matrix or vector, certain rules regarding the dimensions must be followed, ensuring the number of columns in the first matrix matches the number of rows in the second. This compliance is essential for valid operations in matrix algebra, allowing the formation of new matrices through these operations.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors and matrices. It allows us to work with systems of linear equations, which can represent anything from a simple set of equations in mathematics to complex models in engineering and sciences.

A vector, such as \[\vec{x}=\begin{bmatrix}1 \4 \2\end{bmatrix}, \]is an ordered list of numbers that can represent a point in space. In linear algebra, we often use vectors to denote both points and directions in space. These vectors can be added together or scaled by multiplying by a number, an operation called scalar multiplication. However, multiplying a matrix by a vector involves a different kind of calculation.

In our exercise, we calculate the product \(A \vec{x}\) by performing specific row-wise operations. This type of multiplication effectively applies a linear transformation to the vector \(\vec{x}\). You multiply and add corresponding entries of a matrix's row and a vector, giving a new transformed vector. This represents a real-world transformation, like rotating a vector or changing its dimensions, which is often used in graphics and data science for predictions and analysis.

Matrix Operations

Matrix operations include various processes like addition, subtraction, and multiplication. The most common of these is multiplication, particularly matrix-vector multiplication, which was the main focus of our exercise.

To multiply a matrix by a vector, take the rows of the matrix and perform dot products with the vector. This operation involves multiplying each component of a row by the corresponding component in the vector and then summing all the products. Let's break it down:

• First Row: Multiply the first row of matrix \(A\) by vector \(\vec{x}\) to get the first element of the resultant vector. Each entry in the row is multiplied by the corresponding element in the vector.
• Second Row: Do the same for the second row to get the second element of the resultant vector.
• Third Row: Repeat the operation for the third row to find the third element of the new vector.

These calculations yield a new vector, the product \(A \vec{x}=\begin{bmatrix} 8 \7 \3 \end{bmatrix}\).

Matrix operations like these are the building blocks for much more complex calculations in advanced mathematics, computer graphics, and machine learning algorithms, where they perform tasks such as data transformations and object modeling.