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Chapter 2: Q 37E (page 98)

Question:If A is an invertible matrix andcis a nonzero scalar, is the matrixcA invertible? If so, what is the relationship betweenA-1 and( cA )-1?

Short Answer

Expert verified

The matrix is invertible and .cA=鈭�11cA鈭�1

Step by step solution

01

Consider the matrix

If the matrix A is invertible, then, AA鈭�1=I.

As the matrix A is invertible and c鈮�0.

02

Compute the similarity between the matrix and its inverse

Consider the equation.

cA1cA鈭�1=c脳1c脳AA鈭�1=AA鈭�1=I鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�......(1)

Again, consider the equation.

1cA鈭�1cA=c脳1c脳A鈭�1A=A鈭�1A=I鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌�......(2)

Compare the equations (1) and (2).

cA1cA鈭�1=1cA鈭�1cA

Therefore, the matrix cA is invertible and cA=鈭�11cA鈭�1.

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Most popular questions from this chapter

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

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation. 64.

123456X=I2

Question:Consider the matrices of the form A=(abb鈭�a) , where aand bare arbitrary constants. For which values of aand b is A-1= A?

Question: Consider the lines P and Qin R2in the accompanying figure. Consider the linear transformationT(x鈫�)=refQ(refPx鈫�) ; that is, we first reflect x鈫� aboutPand then we reflect the result about Q.

  1. For the vector x鈫�given in the figure, sketch T(x鈫�) . What angle do the vectors x鈫�andT(x鈫�) enclose? What is the relationship between the lengths of x鈫�andT(x鈫�) ?
  2. Use your answer in part (a) to describe the transformation T geometrically, as a reflection, rotation, shear, or projection.
  3. Find the matrix of T .
  4. Give a geometrical interpretation of the linear transformation L(x鈫�)=refP(refQx鈫�), and find the matrix x of L .

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)
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