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Chapter 2: Q13E (page 108)

TRUE OR FALSE?

There exists a real number k such that the matrix [k-23-3k-2]fails to be invertible.

Short Answer

Expert verified

The given statement is false.

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-23-3k-2

The determinant of the matrix is,

detk-23-3k-2=(k-2)(k-2)-3(-3)=k2-4k+13

02

Find the value of unknown

As the matrix fails to be invertible, so,

0=k2-4k+13鈬�k12=-(-4)卤6i2.1鈭�k=2-3i,k=2-3i

There exists the imaginary number and not a real number.

Hence, the given statement is false.

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Most popular questions from this chapter

Consider an n 脳 p matrix A, a p 脳 m matrix B, and a scalar k. Show that (k A)B = A (k B) = k(AB)

Matrixis invertible for all real numbers k.

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]

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).

Write all possible forms of elementary2脳2E5=10k1matrices E. In each case, describe the transformationy鈫�=Ex鈫�geometrically.
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