Section 8.4 Matrix Algebra

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Section 8.4 Matrix Algebra Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding entries of A and B. The difference of A and B, denoted A  B, is obtained by subtracting corresponding entries of A and B. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

The Zero Matrix Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

If A is an m × n matrix and s is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

AB ≠ BA Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

R3=2r1+r3 R1=r1+r2 R3=-4r2+r3 R2=r2 R3=1/9r3 R1=r1+r3 R2=3r3+r2

R1=-1/2r1 R2=4r1+r2 We can see that we cannot get the identity on the left side of the vertical bar. We conclude that A is singular and has no inverse. Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.