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Chapter 9: Problem 40

$$\text {Let } A=\left[\begin{array}{rr} -2 & 4 \\ 0 & 3 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -6 & 2 \\ 4 & 0 \end{array}\right] . \text { Find each of the following.}$$ $$-\frac{3}{2} A$$

Short Answer

Expert verified
\( \left[ 3, -6, 0, -\frac{9}{2} \right] \)

Step by step solution

01

- Understanding the Problem

The problem requires finding the scalar multiplication of matrix A by \(-\frac{3}{2}\). Start by identifying the elements in matrix A to apply the scalar multiplication.
02

- Identify Matrix A

Matrix A is given as: \[ A = \left[ \begin{array}{rr} -2 & 4 \ 0 & 3 \end{array} \right] \]
03

- Multiply Each Element by \(-\frac{3}{2}\)

Multiply each element in matrix A by \(-\frac{3}{2}\): \( -\frac{3}{2} A = -\frac{3}{2} \times \left[ \begin{array}{rr} -2 & 4 \ 0 & 3 \end{array} \right] = \left[ \begin{array}{rr} -\frac{3}{2} \times -2 & -\frac{3}{2} \times 4 \ -\frac{3}{2} \times 0 & -\frac{3}{2} \times 3 \end{array} \right]\)
04

- Calculate the Multiplications

Now, perform the multiplications: \( -\frac{3}{2} \times -2 = 3 \), \(-\frac{3}{2} \times 4 = -6\), \( -\frac{3}{2} \times 0 = 0 \), and \(-\frac{3}{2} \times 3 = -\frac{9}{2}\)
05

- Write the Resulting Matrix

Now, place these results into the matrix format: \( -\frac{3}{2} A = \left[ \begin{array}{rr} 3 & -6 \ 0 & -\frac{9}{2} \end{array} \right] \)

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Key Concepts

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

Matrix Operations
Matrix operations are fundamental in linear algebra and involve actions like addition, subtraction, and multiplication. Each of these operations has specific rules:

- **Addition and Subtraction**: Only matrices of the same dimensions can be added or subtracted. Element-wise addition or subtraction is performed.
- **Multiplication**: There are two types: scalar multiplication and matrix multiplication. Scalar multiplication involves multiplying every element of a matrix by a scalar value. Matrix multiplication involves the dot product of rows and columns from two matrices.

Matrix operations allow us to manipulate arrays of numbers systematically, which is crucial for solving linear equations, transforming geometric shapes, and more.
Matrix Algebra
Matrix algebra encompasses the various rules and techniques used for handling matrices. Among these are:

- **Matrix Addition**: Combine matrices by adding corresponding elements.
- **Matrix Subtraction**: Subtract corresponding elements.
- **Matrix Multiplication**: Combine two matrices but follow a specific rule. The element in row \(i\), column \(j\) of the product matrix is the dot product of row \(i\) of the first matrix and column \(j\) of the second.

- **Transpose**: Switch rows and columns.
- **Inverse**: Find a matrix that, when multiplied with the original, yields the identity matrix.

These operations are fundamental for solving systems of linear equations, performing transformations, and various other algebraic computations. Whether in engineering, computer graphics, or physics, matrix algebra is a powerful tool.

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Most popular questions from this chapter

Supply and Demand In many applications of economics, as the price of an item goes up, demand for the item goes down and supply of the item goes up. The price where supply and demand are equal is the equilibrium price, and the resulting sup. ply or demand is the equilibrium supply or equilibrium demand. Suppose the supply of a product is related to its price by the equation $$p=\frac{2}{3} q$$ where \(p\) is in dollars and \(q\) is supply in appropriate units. (Here, \(q\) stands for quantity.) Furthermore, suppose demand and price for the same product are related by $$p=-\frac{1}{3} q+18$$ where \(p\) is price and \(q\) is demand. The system formed by these two equations has solution \((18,12),\) as seen in the graph. (GRAPH CANNOT COPY) Suppose the price and supply of the can opener are related by \(p=\frac{3}{4} q,\) where \(q\) represents the supply and \(p\) the price. Find the supply at each price. (a) 50 (b) \(\$ 10\) (c) \(\$ 20\)

Your friend missed the lecture on multiplying matrices. Explain to her the process of matrix multiplication.

Use the determinant theorems to find the value of each determinant. $$\left|\begin{array}{rrrr} 4 & 0 & 0 & 2 \\ -1 & 0 & 3 & 0 \\ 2 & 4 & 0 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right|$$

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} &x+3 y=-12\\\ &2 x-y=11 \end{aligned}$$

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product when possible. $$B A$$
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