Review exercise 1.5. For each of the systems of equations in review exercise 1.4, factor the corresponding matrices into the forms or .

Answer: For the first system

the corresponding matrix is

and the final matrix after elimination is

The eliminations steps were as follows:

1. Subtract the first equation from the second equation ().
2. Subtract the first equation from the third equation ().
3. Subtract the second equation from the third equation ().

The matrix of multipliers is thus

and we have the following factorization:

For the second system

the corresponding matrix is

We did an initial exchange of the first and third rows, corresponding to multiplying by the permutation matrix

The final matrix after elimination was

The elimination steps were as follows:

1. Subtract the first equation from the second ().
2. Add the second equation to the third ().

The matrix of multipliers is thus

and we have the following factorization:

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.