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Chapter 2: Q 68E (page 100)

Question: For two invertible n 脳 n matrices Aand B, determine which of the formulas stated in Exercise 67 through 75 are necessarily true.

(A鈭�B)(A+B)=A2鈭�B2

Short Answer

Expert verified

Thus, the given equation is not true.

Step by step solution

01

Use distributive laws for the given matrices.

The given matrices areA andB which is invertible and solve the given equation as follows:

By the distributive laws for matrix multiplication gives

(A-B)(A+B)=(A-B)A+(A-B)B=AA-BA+AB-BB=A2-BA+AB-B2

02

Determine the given equation of matrices is true or false.

Since matrix multiplication is not commutative, BA is usually not equal BA. So the difference AB - BAcannot be written as 0 or cannot be cancelled.

Any matrices Aand B that do not commute because the matrices are of order n 脳 n.

In general, then it gives A鈭�BA+B鈮�A2鈭�B2

Hence, the given equation is not true.

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Most popular questions from this chapter

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?

TRUE OR FALSE?

The formula rref(AB) =rref(A)rref(B)holds for alln脳p matrices A and for allp脳m matrices.

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

123456X=I2

Consider the regular tetrahedron sketched below, whose center is at the

origin.

Let T from 鈩�3to 鈩�3be the rotation about the axis through the points 0 and P2that transforms P1 into P3. Find the images of the four corners of the tetrahedron under this transformation.

P0鈫�rP1鈫�P3P2鈫�P3鈫�

Let L from 鈩�3to 鈩�3be the reflection about the plane through the points 0 , p0andp3. Find the images of the four corners of the tetrahedron under this transformation.

P0鈫�LP1鈫�P2鈫�P3鈫�

Describe the transformations in parts (a) through (c) geometrically.

a.T-1b.L-1c.T2=T鈭�T鈥�(thecompositeofTwithitself)

d. Find the images of the four corners under the transformations T鈭�LandL鈭�T. Are the two transformations the same?

P0鈫�T鈭�LP0鈫�T鈭�LP1鈫�P1鈫�P2鈫�P2鈫�P3鈫�P3鈫�

e. Find the images of the four corners under the transformation L鈭�T鈭�L. Describe this transformation geometrically

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,

[x1x2x3],

Into plane coordinates, [y1y2]. The simplest choice is a linear transformation, for example, the one given by the matrix

[-1210-1201]

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

[000],[100],[010],[001],[110],[011],[101],[111]

Include the images of the axes in your sketch:

(b) Represent the image of the point[11212] in your figure in part (a)

(c.) Find all the points [x1x2x3]in 鈩�3that are transformed to [00]. Explain.
See all solutions

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