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Chapter 2: Q7E (page 85)

If possible, compute the matrix products in Exercises 1 through 13, using paper and pencil.

7.[10-10111-1-2][123221313]

Short Answer

Expert verified

Product of given matrix is.[鈭�110534鈭�6鈭�2鈭�4]

Step by step solution

01

Step1:Matrix multiplication 

If A is matrix of ordern脳pand B is matrix of order.m脳q Then the matrix multiplication ofABIs defined only if.p=m

If Bis am脳qmatrix and An脳pmatrix, then the product BAis defined as the matrix of the linear transformation .T(x鈫�)=B(Ax鈫�)

02

Assuming the matrix

Let the given matrix

A=[10鈭�10111鈭�1鈭�2],B=[123221313]

Order of Matrix Ais3脳3 and order of matrix Bis3脳3.

Since.3=3 Thus the product is defined.

03

Multiplication of matrix 

Now, find the product as follows:

AB=[1.1+0.3+(鈭�1).21.2+0.2+(鈭�1).11.3+0.1+(鈭�1).30.1+1.3+1.20.2+1.2+1.10.3+1.1+3.11.1+(鈭�1).3+(鈭�2).21.2+(鈭�1).2+(鈭�2).11.3+(鈭�1).1+(鈭�2).3]=[鈭�110534鈭�6鈭�2鈭�4]

Hence, the product of given matrix is .[鈭�110534鈭�6鈭�2鈭�4]

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Most popular questions from this chapter

Question:

  1. If Ais any 3x3 transition matrix (see Definition 2.1.4), find the matrix product [111]A .
  2. For a fixed n , let e鈫� be the row vector e鈫�=[1,1,..,1]n1's . Show that an nxn matrix A with nonnegative entries is a transition matrixif (and only if)e鈫�A=e鈫� .

Give a geometric interpretation of the linear transformations defined by the matrices in Exercises 16through 23 . Show the effect of these transformations on the letter L considered in Example 5 . In each case, decide whether the transformation is invertible. Find the inverse if it exists, and interpret it geometrically. See Exercise 13.

21. [01-10]

Find a n脳nmatrix A such thatAx鈫�=3x鈫�for allx鈫�inRn.

Is the product of two lower triangular matrices a lower triangular matrix as well? Explain your answer.

In this exercise we will verify part (b) of Theorem 2.3.11 in the special case when A is the transition matrix [0.40.30.60.7]andx炉is the distribution vector[10]. [We will not be using parts (a) and (c) of Theorem 2.3.11]. The general proof of Theorem 2.3.11 runs along similar lines, as we will see in Chapter 7.

  1. ComputeA[12]andA[1-1]. WriteA[1-1]as a scalar multiple of the vector[1-1].
  2. Write the distribution vectorx鈫�=[10]as a linear combination of the vectors[12]and[1-1]
  3. Use your answers in part (a) and (b) to writeAx鈫�as a linear combination of the vectors[12]and[1-1]. More generally, write Amx鈫�as a linear combination of vectors[12]and[1-1], for any positive integer m. See Exercise 81.
  4. In your equation in part (c), let got to infinity to find limm鈫�鈭�Amx鈫�. Verify that your answer is the equilibrium distribution for A.
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