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Chapter 10: Problem 37

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as follows: $$ A=\left(\begin{array}{rr} 2 & 3 \\ -1 & 4 \end{array}\right) \quad B=\left(\begin{array}{rr} 1 & -1 \\ 3 & 0 \end{array}\right) \quad C=\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right) $$ $$\begin{aligned} &D=\left(\begin{array}{rrr} -1 & 2 & 3 \\ 4 & 0 & 5 \end{array}\right) \quad E=\left(\begin{array}{rr} 2 & 1 \\ 8 & -1 \\ 6 & 5 \end{array}\right)\\\ &F=\left(\begin{array}{rr} 5 & -1 \\ -4 & 0 \\ 2 & 3 \end{array}\right) \quad G=\left(\begin{array}{ll} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array}\right) \end{aligned}$$ In each exercise, carry out the indicated matrix operations if they are defined. If an operation is not defined, say so. $$F D$$

Short Answer

Expert verified
The product matrix is \( \begin{pmatrix} -9 & 10 & 10 \\ 4 & -8 & -12 \\ 10 & 4 & 21 \end{pmatrix} \).

Step by step solution

01

Verify the Order of Matrices

Matrix \( F \) is a \( 3 \times 2 \) matrix and matrix \( D \) is a \( 2 \times 3 \) matrix. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, \( F \) has 2 columns, and \( D \) has 2 rows, so the multiplication is defined.
02

Determine the Order of the Product Matrix

The resulting matrix from the multiplication \( FD \) will have the same number of rows as the first matrix \( F \) and the same number of columns as the second matrix \( D \). Therefore, the product matrix will be \( 3 \times 3 \).
03

Perform Matrix Multiplication

To find the entry in the \( i^{th} \) row and \( j^{th} \) column of the product matrix, calculate the dot product of the \( i^{th} \) row of \( F \) and the \( j^{th} \) column of \( D \). This results in:\(FD = \begin{pmatrix}5(-1) + (-1)4 & 5(2) + (-1)0 & 5(3) + (-1)5 \-4(-1) + 0(4) & -4(2) + 0(0) & -4(3) + 0(5) \2(-1) + 3(4) & 2(2) + 3(0) & 2(3) + 3(5)\end{pmatrix}\)Simplify each entry:- \(5(-1) + (-1)4 = -5 - 4 = -9 \5(2) + (-1)0 = 10 + 0 = 10 \5(3) + (-1)5 = 15 - 5 = 10 \-4(-1) + 0(4) = 4 + 0 = 4 \-4(2) + 0(0) = -8 + 0 = -8 \-4(3) + 0(5) = -12 + 0 = -12 \2(-1) + 3(4) = -2 + 12 = 10 \2(2) + 3(0) = 4 + 0 = 4 \2(3) + 3(5) = 6 + 15 = 21 \)This results in the matrix:\(FD = \begin{pmatrix}-9 & 10 & 10 \4 & -8 & -12 \10 & 4 & 21\end{pmatrix}\)

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Key Concepts

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

Matrix Order
The concept of matrix order, also referred to as dimensions, is crucial in understanding matrix operations. A matrix is essentially a rectangular arrangement of numbers in rows and columns. The **order of a matrix** is defined by its number of rows and columns, typically described as an "m by n" matrix where *m* is the number of rows and *n* is the number of columns.

Understanding matrix order is essential when you are considering operations such as matrix multiplication. For instance, in our example, matrix **F** has an order of \(3 \times 2\), meaning it has 3 rows and 2 columns. Meanwhile, matrix **D** has a \(2 \times 3\) order.

One important rule to remember is that two matrices can only be multiplied if the number of columns in the first matrix is equal to the number of rows in the second matrix. This compatibility is essential for the operation. Thus, having clear knowledge of the order of the matrices at hand helps us identify when operations are defined or undefined, like in our example where **F** and **D** can successfully be multiplied.
Matrix Product
Matrix multiplication, or the **matrix product**, involves combining two matrices to produce a new matrix. When multiplying two matrices, not every pair can be multiplied. This operation demands that the number of columns of the first matrix must be equal to the number of rows of the second matrix.

In our textbook example, **F** and **D** are of orders \(3 \times 2\) and \(2 \times 3\) respectively. Since the number of columns in **F** (which is 2) matches the number of rows in **D** (also 2), matrix multiplication is possible. The resulting product matrix will have an order of the number of rows from the first matrix, **F**, and the number of columns from the second matrix, **D**, thus yielding a \(3 \times 3\) matrix.

Multiplying matrices involves the **dot product**, performed for each element of the new matrix. More on that next.
Dot Product
The process of matrix multiplication heavily relies on the concept of the **dot product**. The dot product helps in determining individual elements of the resulting matrix from a multiplication operation.

For each position in the resulting matrix, you take one entire row from the first matrix and one column from the second matrix, compute their dot product, and place the result in that position. Here鈥檚 how it works:
  • Take the first entry from the row and multiply it by the corresponding first entry from the column.
  • Do this for all entries in the row and column.
  • Add up these products to get the final number for that position.
In our example for the position in the first row and first column of matrix **FD**, we calculated it as: \(5(-1) + (-1)4 = -5 - 4 = -9\).

This "dot product" calculation is performed for each element of the resulting matrix. This method might seem complex but breaks down the multiplication into manageable, repeatable steps.

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Most popular questions from this chapter

A sketch shows that the line \(y=100 x\) intersects the parabola \(y=x^{2}\) at the origin. Are there any other intersection points? If so, find them. If not, explain why not.

In this exercise, we continue to explore some of the connections between matrices and geometry. As in Exercise \(59,\) we will use \(2 \times 1\) matrices to specify the coordinates of points in the plane. Let \(P, S,\) and \(T\) be the matrices defined as follows: $$ \begin{array}{c} P=\left(\begin{array}{c} \cos x \\ \sin x \end{array}\right) \quad S=\left(\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right) \\ T=\left(\begin{array}{cc} \cos \beta & -\sin \beta \\ \sin \beta & \cos \beta \end{array}\right) \end{array} $$ Notice that the point \(P\) lies on the unit circle. (a) Compute the matrix \(S P .\) After computing \(S P,\) observe that it represents the point on the unit circle obtained by rotating \(P\) (about the origin) through an angle \(\theta\) (b) Show that $$ S T=T S=\left(\begin{array}{cc} \cos (\theta+\beta) & -\sin (\theta+\beta) \\ \sin (\theta+\beta) & \cos (\theta+\beta) \end{array}\right) $$ (c) Compose (ST)P? What is the angle through which \(P\) is rotated?

(a) Compute the inverse of the coefficient matrix for the system. (b) Use the inverse matrix to solve the system. In cases in which the final answer involves decimals, round to three decimal places. $$\left\\{\begin{array}{cc} 5 x-2 y-2 z & =15 \\ 3 x+y & =4 \\ x+y+z & =-4 \end{array}\right.$$

So the input (3,5) yields an output of \(\sqrt{2} .\) We define the do- main for this function just as we did in Chapter 3: The domain is the set of all inputs that yield real-number outputs. For instance, the ordered pair (1,4) is not in the domain of the function we have been discussing, because (as you should check for yourself\() f(1,4)=\sqrt{-1},\) which is not a real number. We can determine the domain of the function in equation ( 1 ) by requiring that the quantity under the radical sign be non negative. Thus we require that \(2 x-y+1 \geq 0\) and, consequently, \(y \leq 2 x+1\) (Check this.) The following figure shows the graph of this inequality; the domain of our function is the set of ordered pairs making up the graph. In Exercises follow a similar procedure and sketch the domain of the given function. (Graph cant copy) $$h(x, y)=\sqrt{x}+\sqrt{y}$$

Show that $$\left|\begin{array}{llll} a & 1 & 1 & 1 \\ 1 & a & 1 & 1 \\ 1 & 1 & a & 1 \\ 1 & 1 & 1 & a \end{array}\right|=(a-1)^{3}(a+3)$$
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