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Chapter 9: Problem 45

Suppose that matrix \(A\) has dimension \(2 \times 3, B\) has dimension \(3 \times 5,\) and \(C\) has dimension \(5 \times 2 .\) Decide whether the given product can be calculated. If it can, determine its dimension. $$4 B$$

Short Answer

Expert verified
Yes, the product can be calculated. The resulting matrix will have dimension 3x5.

Step by step solution

01

Identify the Dimension of Matrix B

Matrix B has dimensions of 3 rows by 5 columns, which is written as a 3x5 matrix.
02

Understand the Scalar Multiplication

When multiplying a matrix by a scalar, every element in the matrix is multiplied by that scalar. The dimensions of the matrix remain unchanged.
03

Determine the Dimension of the Product

Since multiplying matrix B by the scalar 4 does not change the dimensions of matrix B, the resulting matrix will also have dimensions of 3 rows by 5 columns, or 3x5.

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Key Concepts

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

scalar multiplication
Scalar multiplication is an essential operation in linear algebra. It involves multiplying each element of a matrix by a scalar value. For example, if we have a scalar 4 and a matrix B with elements \( b_{ij} \), scalar multiplication would mean computing \( 4 \times b_{ij} \) for every element in B. This operation is straightforward but crucial, as it forms the basis for more complex matrix operations. The most important takeaway is that scalar multiplication doesn't alter the dimensions of the matrix. In our case, multiplying matrix B by 4 results in matrix B having the same dimensions, 3 rows and 5 columns.
matrix dimensions
Understanding the dimensions of matrices is critical for performing any matrix operation. A matrix's dimensions are described as \( m \times n \), where m represents the number of rows and n represents the number of columns.
For example, a matrix B with dimensions 3 \( \times \) 5 has 3 rows and 5 columns. Knowing the dimensions aids in checking compatibility for various operations like addition, subtraction, and multiplication.
Proper knowledge of matrix dimensions ensures that operations are performed correctly, avoiding errors and making problem-solving more efficient.
matrix operations
Matrix operations include addition, subtraction, multiplication, and scalar multiplication. Here are brief explanations:
  • Addition: Matrices must have the same dimensions. Corresponding elements are added together.
  • Subtraction: Similar to addition, but corresponding elements are subtracted.
  • Multiplication: More complex. For two matrices A and B to be multiplied, the number of columns in A must equal the number of rows in B. The resulting matrix's dimensions will be determined by the rows of A and the columns of B.
  • Scalar Multiplication: As discussed earlier, every element in the matrix is multiplied by a scalar value.
Efficient understanding and execution of these operations are foundational to exploring more sophisticated concepts in linear algebra.

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Most popular questions from this chapter

Gasoline Revenues The manufacturing process requires that oil refineries manufacture at least 2 gal of gasoline for each gallon of fuel oil. To meet the winter demand for fuel oil, at least 3 million gal per day must be produced. The demand for gasoline is no more than 6.4 million gal per day. If the price of gasoline is \(\$ 2.90\) per gal and the price of fuel oil is \(\$ 2.50\) per gal, how much of each should be produced to maximize revenue?

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. $$C B$$

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} 3 x+4 y &=-3 \\ -5 x+8 y &=16 \end{aligned}$$

Solve each problem. Purchasing costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix \(A\) Eggs lour Sugar Shortening Milk \(\left.\begin{array}{l|ccccc}\text { Rolls (doz) } & 1 & 4 & \frac{1}{4} & \frac{1}{4} & 1 \\ \text { Bread (loaf) } & 0 & 3 & 0 & \frac{1}{4} & 0 \\ \text { Cake } & 4 & 3 & 2 & 1 & 1 \\ \text { Pie (crust) } & 0 & 1 & 0 & \frac{1}{3} & 0\end{array}\right]=A\) The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix \(B\) Large Lot Small Lot \(\left.\begin{array}{l|rr}\text { Eggs } & 5 & 5 \\ \text { Flour } & 8 & 10 \\\ \text { Sugar } & 10 & 12 \\ \text { Shortening } & 12 & 15 \\ \text { Milk } & 5 & 6\end{array}\right]=B\) (a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options. (b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a \(1 \times 4\) matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders. (c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to determine the solution set. $$\begin{array}{l} 2 x-3 y+z-8=0 \\ -x-5 y+z+4=0 \\ 3 x-5 y+2 z-12=0 \end{array}$$
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