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Chapter 2: Problem 4

Perform the indicated operations when \(a=3\) \(b=-4,\) and $$A=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right], \quad B=\left[\begin{array}{rr} 0 & 1 \\ -1 & 2 \end{array}\right], \quad O=\left[\begin{array}{ll} 0 & 0 \\ 0 & 0 \end{array}\right]$$ $$(a+b) B$$

Short Answer

Expert verified
The result of the operation \((a+b) B\) is the matrix \[\left[\begin{array}{rr} 0 & -1 \ 1 & -2 \end{array}\right]\].

Step by step solution

01

Compute the sum of a and b

The first step is to sum the values of a and b. Here, \(a=3\) and \(b=-4\). Hence, \(a+b = 3+(-4) = -1\).
02

Perform Scalar Multiplication

Next, multiply the resulting scalar value from step 1 with matrix B. Scalar multiplication is done element-by-element such that each entry of the matrix is multiplied by the scalar value. So, if we represent the multiplication as (-1)*B, it produces a new matrix: \[C =\left[\begin{array}{rr} 0*(-1) & 1*(-1) \ -1*(-1) & 2*(-1) \end{array}\right] = \left[\begin{array}{rr} 0 & -1 \ 1 & -2 \end{array}\right]\].
03

Result

The result of performing \((a+b) B\) is matrix C which is \[\left[\begin{array}{rr} 0 & -1 \ 1 & -2 \end{array}\right]\].

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Key Concepts

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

Matrix Arithmetic
Matrix arithmetic is fundamental to linear algebra and involves operations that can be performed on matrices, such as addition, subtraction, and multiplication. A matrix is a rectangular array of numbers arranged in rows and columns, and it's essential to understand the rules governing these operations to effectively manipulate matrices.

When adding or subtracting matrices, it's important to note that you can only combine matrices of the same dimensions by adding or subtracting corresponding elements. For instance, consider two matrices, A and B, both of which are 2x2 matrices. Their sum, A + B, is formed by adding each element in A to the matching element in B. The same principle applies to subtraction.

Matrix multiplication, on the other hand, has more intricate rules and involves the dot product of the rows and columns of the matrices being multiplied. Notably, if the number of columns in the first matrix does not match the number of rows in the second matrix, the matrices cannot be multiplied together. Understanding these rules is crucial to carrying out more complex operations in linear algebra.
Scalar Multiplication
Scalar multiplication is a linear algebra operation where every element in a matrix is multiplied by a single number, known as a scalar. This scalar is not a matrix but a real number and when multiplied with a matrix, does not change the size of the matrix, only the magnitude of each element is adjusted.

For example, if the scalar is 2 and the matrix is A, in scalar multiplication you would multiply every element in A by 2. The result is a new matrix, where each element is twice as large as the corresponding element in A. Scalar multiplication is associated with concepts like dilation and contraction in geometry because it can either expand or shrink a matrix's elements if the scalar is greater or less than one, respectively.

Scalar multiplication is important in real-world applications such as scaling graphics in computer graphics, adjusting the strength of forces in physics, and updating predictions in data models.
Linear Algebra Operations
Linear algebra involves a variety of operations that can be applied to matrices and vectors, which are an integral part of fields such as computer science, engineering, physics, and more. The fundamental operations in linear algebra include matrix addition, matrix multiplication, scalar multiplication, matrix inversion, and transposition.

Understanding Linear Transformations

Linear transformations are mappings from one vector space to another that preserve the operations of vector addition and scalar multiplication. They can be represented by matrices, making matrix operations crucial in understanding and computing transformations.

Utilizing Matrices for Systems of Equations

One of the primary uses of matrices in linear algebra is to solve systems of linear equations. Representing a system in a matrix form enables the use of various algorithms for finding solutions efficiently.

Overall, mastering these operations allows for the exploration of more advanced topics like eigenvectors and eigenvalues, which have applications in areas such as dynamic systems, quantum mechanics, and machine learning.

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Most popular questions from this chapter

find the inverse of the elementary matrix. $$\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -3 & 1 \end{array}\right]$$

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