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Chapter 2: Q12 E (page 108)

TRUE OR FALSE?

There exists a real number k such that the matrix[k-1-2-4k-3] fails to be invertible.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Compute the determinant of the matrix.

The matrix is said to be invertible if the determinant of that matrix is nonzero.

Consider the matrix.

k-1-2-4k-3

The determinant of the matrix is,

detk-1-2-4k-3=k-1k-3--2-4=k2-4k-5

As the matrix fails to be invertible, so,

0=k2-4k-5鈬�(k-5)(k+1)=0鈭�k=-1,5

There exists the real number.

02

Find the value of unknown

As the matrix fails to be invertible, so,

0=k2-4k-5鈬�(k-5)(k+1)=0鈭�k=-1,5

There exists the real number.

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Most popular questions from this chapter

Consider two n x nmatrices A and B whose entries are positive or zero. Suppose that all entries of A are less than or equal to 鈥�s鈥�, and all column sums of B are less than or equal to 鈥�r鈥� (the column sum of a matrix is the sum of all the entries in its column). Show that all entries of the matrix AB are less than or equal to 鈥�sr鈥�.

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.

TRUE OR FALSE?

There exists a matrix A such that[1234]A[5678]=[1111] .

Which of the functionsf fromR toR in Exercises 21 through 24 are invertible?23 f(x)=x3+x.

TRUE OR FALSE?

The formula rref(AB) =rref(A)rref(B)holds for alln脳p matrices A and for allp脳m matrices.

See all solutions

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