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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

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 . $(c A) {}^{鈭� 1}= \frac{1}{c} A^{鈭� 1}$

Step by step solution

01

Consider the matrix

If the matrix A is invertible, then, $A A^{鈭� 1} = I$ .

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

02

Compute the similarity between the matrix and its inverse

Consider the equation.

$\begin{matrix} (c A) (\frac{1}{c} A^{鈭� 1}) = c 脳 \frac{1}{c} 脳 A A^{鈭� 1} \\ = A A^{鈭� 1} \\ = I 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� ...... (1) \end{matrix}$

Again, consider the equation.

$\begin{matrix} (\frac{1}{c} A^{鈭� 1}) (c A) = c 脳 \frac{1}{c} 脳 A^{鈭� 1} A \\ = A^{鈭� 1} A \\ = I 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� ...... (2) \end{matrix}$

Compare the equations (1) and (2).

$(c A) (\frac{1}{c} A^{鈭� 1}) = (\frac{1}{c} A^{鈭� 1}) (c A)$

Therefore, the matrix cA is invertible and $(c A) {}^{鈭� 1}= \frac{1}{c} A^{鈭� 1}$ .

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Most popular questions from this chapter

One of the five given matrices represents an orthogonal projection onto a line and another represents a reflection about a line. Identify both and briefly justify your choice.

$\begin{matrix} A = \frac{1}{3} [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}], 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� B = \frac{1}{3} [\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}], \\ C = \frac{1}{3} [\begin{matrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{matrix}], 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆€夆€� D = 鈭� \frac{1}{3} [\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{matrix}], \\ E = \frac{1}{3} [\begin{matrix} 鈭� 1 & 2 & 2 \\ 2 & 鈭� 1 & 2 \\ 2 & 2 & 鈭� 1 \end{matrix}] \end{matrix}$

Consider a linear transformation Tfrom $R^{2}$ to $R^{2}$ . Suppose that $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ are two arbitrary vectors in $R^{2}$ and that $\overset{鈫�}{x}$ is a third vector whose endpoint is on the line segment connecting the endpoints of $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ . Is the endpoint of the vector $T (\overset{鈫�}{x)}$ necessarily on the line segment connecting the endpoints of $T (\overset{鈫�}{v)}$ and $T (\overset{鈫�}{w)}$ ? Justify your answer.

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the orthogonal projection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ onto line 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)

Consider the transformation T from $R^{2}$ to $R^{2}$ that rotates any vector $\overset{鈫�}{x}$ through an angle of $45 掳$ in the counterclockwise direction, as shown in the following figure:

You are told that T is a linear transformation. (This will be shown in the next section.) Find the matrix of T .

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