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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: Q54E (page 109)

TRUE OR FALSE?
The formula rref(AB) =rref(A)rref(B)holds for all $n 脳 p$ matrices A and for all $p 脳 m$ matrices.

Short Answer

Expert verified

The given statement is true.

Step by step solution

01

Reduce row echelon form

The reduced row echelon form of a matrix contains leading ones followed by zeros in each row with a shift in place of leading ones with every next row.

02

Consider the parameters

The product of rref of matrices will result in either identity matrix or the identity matrix followed by rows of zeros.

Thus, the formula rref(AB) = rref (A) rref(B) holds good.

Hence, the statement is true.

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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.

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

When you represent a three 鈥揹imensional object graphically in the plane (on paper, the blackboard, or a computer screen), you have to transform spatial coordinates,

$[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}]$ ,

Into plane coordinates, $[\begin{matrix} y_{1} \\ y_{2} \end{matrix}]$ . The simplest choice is a linear transformation, for example, the one given by the matrix

$[\begin{matrix} - \frac{1}{2} & 1 & 0 \\ - \frac{1}{2} & 0 & 1 \end{matrix}]$

(a.) Use this transformation to represent the unit cube with corner points

$[\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$

Include the images of the axes in your sketch:

(b) Represent the image of the point $[\begin{matrix} 1 \\ \frac{1}{2} \\ \frac{1}{2} \end{matrix}]$ in your figure in part (a)

(c.) Find all the points $[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}]$ in ${鈩�}^{3}$ that are transformed to $[\begin{matrix} 0 \\ 0 \end{matrix}]$ . Explain.

The trace of a matrix $[\begin{matrix} a & b \\ c & d \end{matrix}]$ is the sum $a + d$ of its diagonal entries. What can you say about the trace of a localid="1664169930199" $2 脳 2$ matrixthat represents localid="1664169941875" $a (n)$ .

a. orthogonal projection b. reflection about a line

c. rotation d. (horizontal or vertical) shear.

In three cases, give the exact value of the trace, and in one case, give an interval of possible values.

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

Use the given partitions to compute the products in Exercises 15 and 16. Check your work by computing the same products without using a partition. Show all your work.

16. $[\begin{matrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} 1 & 2 & 2 & 3 \\ 3 & 4 & 4 & 5 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 3 & 4 \end{matrix}]$

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