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Chapter 2: Q30E (page 98)

For which values of the constants b and c is the following matrix invertible?

[01b-10-c-b-c0]

Short Answer

Expert verified

The matrix[01b-10-c-b-c0] is not invertible for any value of b and c.

Step by step solution

01

  Definition of an invertible matrix

A matrix A of order n x n is called an invertible matrix if (and only if) the determinant of matrix A is equal to zero.

i.e. A=0

02

  Finding the determinant of the given matrix

LetA=[01b-10-c-b-c0]

Find the determinant as:

A=00-c-10-bc+bc-0A=0+bc-bcA=0

Since the determinant of the matrix A is equal to zero. Therefore, the matrix A is not invertible for any value of b and c.

03

  Final Answer

The matrix[01b-10-c-b-c0] is not invertible for any value of b and c.

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Most popular questions from this chapter

Which of the functionsf fromR toR in Exercises 21 through 24 are invertible?21 f(x)=x2.

Let A=[110-312]in all parts of this problem.

(a) Find the scalar位 such that the matrixA-位濒2fails to be invertible. There are two solutions; choose one and use it in parts (b) and (c).

(b) For the 位you choose in part (a), find a non-zero vector x鈬赌such that

role="math" localid="1659697491583" (A-位濒2)x鈬赌=0鈬赌

(c) Note that the equation (A-位濒2)x鈬赌=0鈬赌can be written as.

A-位x鈬赌=0鈬赌orAx鈬赌=位x鈬赌Check that the equationAx鈬赌=位x鈬赌 holds for your位from part (a) and yourx鈬赌from part (b).

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�.

There exists a nonzero upper triangular 2 脳 2 matrix A such that A2=[0000].

Consider a linear transformation Tfrom R2to R2. Suppose thatv鈫�and w鈫� are two arbitrary vectors in R2and thatx鈫�is a third vector whose endpoint is on the line segment connecting the endpoints ofv鈫�and w鈫�. Is the endpoint of the vectorT(x)鈫�necessarily on the line segment connecting the endpoints ofT(v)鈫�and T(w)鈫�? Justify your answer.
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