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Chapter 1: Problem 1

In Exercises 1-i6, let $$ A=\left[\begin{array}{rrr} -2 & 1 & 3 \\ 4 & 0 & -1 \end{array}\right], \quad B=\left[\begin{array}{rrr} 4 & 1 & -2 \\ 5 & -1 & 3 \end{array}\right], \quad C=\left[\begin{array}{rr} 2 & -1 \\ 0 & 6 \\ -3 & 2 \end{array}\right], \quad \text { and } \quad D=\left[\begin{array}{rr} -4 & 2 \\ 3 & 5 \\ -1 & -3 \end{array}\right] \text {. } $$ Compute the indicated quantity, if it is defined. $$ 3 A $$

Short Answer

Expert verified
The matrix \(3A\) is \(\begin{bmatrix}-6 & 3 & 9 \\ 12 & 0 & -3\end{bmatrix}\).

Step by step solution

01

Understand the Problem

The exercise asks to compute \(3A\), where \(A\) is a given matrix. This involves scalar multiplication of the matrix \(A\) by the scalar 3. Scalar multiplication involves multiplying each entry of the matrix by the scalar.
02

Setup for Scalar Multiplication

Write the matrix \(A\) which is given as: \[A=\begin{bmatrix}-2 & 1 & 3 \ 4 & 0 & -1\end{bmatrix}\]Prepare to multiply each element of \(A\) by 3.
03

Perform the Scalar Multiplication

Multiply each element of the matrix \(A\) by 3:- The element in the first row, first column: \(-2 \times 3 = -6\)- The element in the first row, second column: \(1 \times 3 = 3\)- The element in the first row, third column: \(3 \times 3 = 9\)- The element in the second row, first column: \(4 \times 3 = 12\)- The element in the second row, second column: \(0 \times 3 = 0\)- The element in the second row, third column: \(-1 \times 3 = -3\)
04

Write the Resulting Matrix

After performing the scalar multiplication, the resulting matrix \(3A\) is:\[3A = \begin{bmatrix}-6 & 3 & 9 \ 12 & 0 & -3\end{bmatrix}\]

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Key Concepts

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

Matrix Arithmetic
Matrix arithmetic refers to the operations that can be carried out on matrices, such as addition, subtraction, and multiplication. These operations follow specific rules similar to arithmetic for numbers but are designed to work within the framework of matrices.
One key operation is **scalar multiplication**. This involves multiplying every element of a matrix by a scalar value. The operation results in each element of the matrix being scaled up or down depending on the scalar. For example, when computing \(3A\), a matrix \(A\) is multiplied element-wise by the scalar 3.

Matrix arithmetic can also include:
  • **Matrix Addition:** Adding corresponding elements from two matrices of the same dimensions.
  • **Matrix Subtraction:** Subtracting corresponding elements from one matrix from another matrix of the same dimensions.
  • **Matrix Multiplication:** Multiplying rows by columns, which contributes to more complex operations like finding the product of two matrices.
Understanding these operations is critical because they form the basis for more advanced matrix techniques in mathematics, science, and engineering.
Linear Algebra
Linear algebra is a branch of mathematics that is primarily concerned with linear equations, linear sets of equations, and their representations through matrices and vector spaces. Matrices serve as a fundamental tool in linear algebra because they can represent systems of linear equations succinctly.

**Matrices: Building Blocks of Linear Algebra**
Matrices are rectangular arrays of numbers arranged in rows and columns. They can represent transformations from one vector space to another, leveraging operations such as matrix addition and multiplication, which simplify the process of solving systems of linear equations.

**Scalars in Linear Algebra**
  • Scalars are real numbers that are used for scaling vectors and matrices.
  • Scalar multiplication, such as in the exercise \(3A\), is a foundational operation in linear algebra. It keeps the matrix's dimensions the same, only changing the magnitude of its elements.
Linear algebra is widely used in various scientific fields including physics, computer science, economics, and statistics due to its ability to model complex systems in a manageable way.
Matrix Operations
Matrix operations are the procedures you can perform on matrices to solve equations or to apply matrix functions. These operations include things like addition, multiplication, and finding determinants.

**Scalar Multiplication in Matrix Operations**
  • This operation involves multiplying every entry of a matrix by a fixed scalar value.
  • It will only change the individual elements, not the overall structure or size of the matrix.
  • This operation is commutative within itself; meaning \(c(A) = A(c)\), where \(c\) is a scalar and \(A\) is the matrix.
**Other Operations in Matrix Arithmetic**
  • **Matrix Transposition:** Flipping a matrix over its diagonal, changing the rows to columns.
  • **Matrix Inversion:** Finding the inverse of a square matrix, which when multiplied to the original yields the identity matrix.
Matrix operations allow complex transformations and manipulations of data, demonstrating their importance in not just mathematics but practical applications across different fields.

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Most popular questions from this chapter

Compute \(\mathbf{v}+\mathbf{w}\) and \(\mathbf{v}-\mathbf{w}\) for the given vectors \(\mathbf{v}\) and \(\mathbf{w}\). Then draw coordinate axes and sketch, using your answers, the vectors \(\mathrm{v}\), \(\mathrm{w}, \mathrm{v}+\mathrm{w}\), and \(\mathrm{v}-\mathrm{w}\). $$v=[2,-1], w=[-3,-2]$$

In Exercises 1-6, reduce the matrix to (a) row-echelon form, and (b) reduced row-echelor: form. Answers to \((a)\) are not unique, so your answer may differ from the one at the back of the text. \(\left[\begin{array}{rrrrr}-1 & 3 & 0 & 1 & 4 \\ 1 & -3 & 0 & 0 & -1 \\ 2 & -6 & 2 & 4 & 0 \\ 0 & 0 & 1 & 3 & -4\end{array}\right]\)

In Exercises \(1-17\), let \(\mathrm{u}=[-1,3,4], \mathrm{v}=\) \([2,1,-1]\), aid \(w=[-2,-1,3]\). Find the indicated quantity. A nonzero vector perpendicular to both \(u\) and \(\mathbf{w}\)

In Exercises \(1-17\), let \(\mathrm{u}=[-1,3,4], \mathrm{v}=\) \([2,1,-1]\), aid \(w=[-2,-1,3]\). Find the indicated quantity. A nonzero vector perpendicular to both u and \(v\)

Mark each of the following True or False. a. Every linear system with the same number of equations as unknowns has a unique solution. b. Every linear system with the same number of equations as unknowns has at least one solution. c. A linear system with more equations than unknowns may have an infinite number of solutions. d. A linear system with fewer equations than unknowns may have no solution. e. Every matrix is row equivalent to a unique matrix in row-echelon form. f. Every matrix is row equivalent to a unique matrix in reduced row-echelon form. g. If \([A \mid\) b \(]\) and \([B \mid\) c \(]\) are row-equivalent partitioned matrices, the linear systems \(A \mathrm{x}=\mathrm{b}\) and \(B \mathrm{x}=\mathrm{c}\) have the same solution set. h. A linear system with a square coefficient matrix \(A\) has a unique solution if and only if \(A\) is row equivalent to the identity matrix. i. A linear system with coefficient matrix \(A\) has an infinite number of solutions if and only if \(A\) can be row-reduced to an echelon matrix that includes some column containing no pivot. j. A consistent linear system with coefficient matrix \(A\) has an infinite number of solutions if and only if \(A\) can be row-reduced to an echelon matrix that includes some column containing no pivot.
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