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Chapter 6: Q7E (page 308)

det(A+B)=detA+detBfor all5脳5 matricesA andB.

Short Answer

Expert verified

Therefore, the given condition is not satisfied. So, the given statement is false.

Step by step solution

01

Definition. 

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

02

To check whether the given condition is true or false. 

Let,

A=1000001000001000001000001

B=0000000000000000000000001

It applies

detA=detB=0,

But

det(A+B)=detI5det(A+B)=1

Therefore, the given condition is not satisfied. So, the given statement is false.

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Most popular questions from this chapter

Consider two distinct real numbers, a and b. We define the function

f(t)=det[111abta2b2t2]

a. Show thatf(t) is a quadratic function. What is the coefficient oft2?
b. Explain whyf(a)=f(b)=0. Conclude thatf(t)=k(t-a)(t-b), for some constant k. Find k, using your work in part (a).
c. For which values of tis the matrix invertible?

Show that the function

F[abcdefghj]=bfg

is linear in all three columns and in all three rows. See Example 6. Is F alternating on the columns? See Example 4.

In Exercises 62 through 64, consider a function D from 鈩�2脳2 to鈩� that is linear in both columns and alternating on the columns. See Examples 4 and 6 and the subsequent discussions. Assume thatD(l2)=1.


62. Show thatD(A)=0for any2脳2matrix whose two columns are equal.

If all the diagonal entries of an n脳nmatrix Aare odd integers and all the other entries are even integers, thenA must be an invertible matrix.

If all the columns of a square matrixAare unit vectors, then the determinant ofAmust be less than or equal to 1.
See all solutions

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