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Chapter 3: Q1P (page 140)

Use index notation as in (9.9) to prove the second part of the associative law for matrix multiplication: (AB)C = ABC

Short Answer

Expert verified

The second part of associative law for matrix is proved by showing that (AB)Cij=ABCij.

Step by step solution

01

Multiplication of Matrices

The element in row i and column j of the product matrix AB is equal to i row of A times column j of B . In index notation, (AB)ij=∑AikBkjk

02

Prove second part of associative law

The index notation is to be used to prove the second part of associative law of matrix multiplication.

Take as the product and as matrix and use the index notation for matrix multiplication to state that(MN)ij=∑kMikNkj.

Substitute M = AB and N = C .

(AB)Cij=∑kABikCkj

Again, use the index notation for the multiplication of and.

ABik=AilBik

Therefore,ABCcan be simplified as follows:

ABCij=∑kABikCkj=∑k∑lAilBikCkj=∑k∑lAilBikCkj=ABCij

Hence, the second part of associative law for matrix,ABC=ABC , has been proved.

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Most popular questions from this chapter

For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using.

6.x+y-z=13x+2y-2z=3

In Problems8to15,use(8.5)to show that the given functions are linearly independent.

13.sin⁡x,sin⁡2x

Find the eigenvalues and eigenvectors of the real symmetric matrix

M=(AHHB)

Show that the eigenvalues are real and the eigenvectors are perpendicular.

In Problems 8to15,use (8.15) to show that the given functions are linearly independent.

sinx,cosx,xsinx,xcosx.

Draw diagrams and prove (4.1).
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