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Chapter 2: Q 67E (page 100)

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

(A+B)2=A2+2AB+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 are and which is invertible and solve the given equation as follows:

By the distributive laws for matrix multiplication gives,

(A+B)2=(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 sum AB + BA can not be written as 2AB.

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

In general, then it gives A+B2鈮�A2+2AB+B2

Hence, the given equation is not true.

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Most popular questions from this chapter

Consider two n x nmatrices A and B whose entries are positive or zero. Suppose that all entries of A are less than or equal to 鈥�s鈥�, and all column sums of B are less than or equal to 鈥�r鈥� (the column sum of a matrix is the sum of all the entries in its column). Show that all entries of the matrix AB are less than or equal to 鈥�sr鈥�.

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