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Chapter 33: Problem 781

Given the matrix \(A=\mid 1 \quad 2 \quad 3\) and the column ve \(\mathrm{x}=|9|\) \(|-1|\) \(|-2|\) 101 find the product \(A X\).

Short Answer

Expert verified
The product AX is equal to 1.

Step by step solution

01

Identify the matrices A and X

We are given the row matrix A and column matrix X: Matrix A = \(\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}\) Matrix X = \(\begin{pmatrix} 9 \\ -1 \\ -2 \end{pmatrix}\)
02

Set up the matrix multiplication

To find the product AX, we should multiply the row matrix A with the column matrix X. It will result in a single-element matrix, i.e., a scalar value. Matrix A has 1 row and 3 columns, while matrix X has 3 rows and 1 column.
03

Multiplying the matrices

To multiply the row matrix A with the column matrix X, we need to perform the following operation: AX = \(\begin{pmatrix} 1 & 2 & 3 \end{pmatrix}\) 脳 \(\begin{pmatrix} 9 \\ -1 \\ -2 \end{pmatrix}\) As A has 1 row and 3 columns, and X has 3 rows and 1 column, their product will be a 1x1 matrix array (a scalar value). Following the rules of matrix multiplication, the product AX can be calculated by multiplying the elements of the row in A by the corresponding elements of the column in X and summing the results: AX = (1 脳 9) + (2 脳 -1) + (3 脳 -2)
04

Calculate the final result

Perform the required arithmetic operations as identified in step 3: AX = (9) + (-2) + (-6) AX = 9 - 2 - 6 AX = 1 Thus, the product AX = 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Row Matrix
In linear algebra, a row matrix is a type of matrix that consists of a single row of elements. For example, consider matrix \( A = \begin{pmatrix} 1 & 2 & 3 \end{pmatrix} \). This is a row matrix because it contains one row with multiple columns. Row matrices are denoted by the arrangement where all elements are positioned horizontally.
  • Properties: A row matrix is defined as a \( 1 imes n \) matrix, where \( n \) is the number of columns.
  • Application: Row matrices are often used in linear equations, transformations, and to perform matrix operations such as multiplication with other matrices.
When multiplying a row matrix by a column matrix, the result is often a single value, called a scalar value. This operation is integral to understanding how to manipulate matrices in various mathematical and applied contexts.
Column Matrix
A column matrix, unlike a row matrix, consists of a single column of elements. For example, consider \( X = \begin{pmatrix} 9 \ -1 \ -2 \end{pmatrix} \). In this matrix, each element sits in its own row, but they are all in one column.
  • Properties: A column matrix is characterized by its \( m \times 1 \) dimension, with \( m \) representing the number of rows.
  • Usage: Column matrices are prevalent in vector mathematics and are used to hold data, represent variables, or facilitate vector operations.
The interaction between row and column matrices is fundamental in matrix multiplication and is the basis for obtaining results such as scalar values when multiplying a single row by a column.
Scalar Value
A scalar value is a single number that results from certain operations on matrices or vectors. In our matrix multiplication example, when the row matrix \( A \) is multiplied by the column matrix \( X \), the result is a single scalar value. The calculation went as follows:
AX = (1 脳 9) + (2 脳 -1) + (3 脳 -2)
  • Meaning: A scalar is often viewed as a \( 1 \times 1 \) matrix, or simply a single numerical value.
  • Importance: Scalars are crucial in scaling operations, where each element of a matrix may be multiplied by a scalar to transform its scale.
The product of a row matrix and a column matrix, resulting in a scalar, highlights the power of matrix operations in simplifying complex calculations.
Matrix Product
The concept of the matrix product is central to understanding many calculations in linear algebra. When we multiply two matrices, say a row matrix and a column matrix, the operation is known as a matrix product.
  • Steps: Identify matrices, verify dimensions are compatible for multiplication, multiply corresponding elements and sum up these products.
  • Compatibility: Matrix A \( 1 \times n \) can be multiplied by matrix X \( n \times 1 \) because the number of columns in A matches the number of rows in X.
The matrix product \( AX \) resulted in a scalar, illustrating how specific configurations in matrix dimensions yield different types of results, ranging from new matrices to singular scalar values. This operation is essential for applications in areas such as physics, computer graphics, and more.

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Most popular questions from this chapter

Find the eigenvalues of: $$ \begin{array}{rlrr} A= & \mid \begin{array}{rrrr} 4 & 1 & 0 \end{array} \\ & \mid-1 & 2 & 0 \\ & \mid 2 & 1 & -3 \mid \end{array} $$

Put the following system into matrix form: $$ \begin{aligned} &(\mathrm{dx} / \mathrm{dt})=\mathrm{x}+\mathrm{y} \\ &(\mathrm{dy} / \mathrm{dt})=9 \mathrm{x}+\mathrm{y} \end{aligned} $$

Solve the system of second-order differential equations $$ \left(\mathrm{D}^{2}-1\right) \mathrm{x}+\mathrm{D}^{2} \mathrm{y}=-2 \sin \mathrm{t} $$ (a) $$ \left(D^{2}+1\right) x+D^{2} y=0 $$ (b)

Verify the Cayley-Hamilton theorem for the matrix $$ \begin{array}{rrr} \mid 4 & 1 & 0 \mid \\ \mid-1 & 2 & 0 \mid \\ 12 & 1 & -3 \mid \end{array} $$

Express the differential equation \(\ddot{\mathrm{x}}-6 \dot{\mathrm{x}}+9 \mathrm{x}=\mathrm{t}\) as a first-order matrix system $$ \dot{x}(t)=A(t) x(t)+f(t) $$
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