Author Archives: hecker

Review exercise 1.8. Given the following matrices: for a matrix how are the rows of related to the rows of ? Answer: For the first matrix , the product is a 3 by 3 matrix in which: The first row … Continue reading →

Review exercise 1.7. For each of the each of the 2 by 2 matrices containing only -1 or 1 as entries, determine whether the matrix is invertible or not. Answer: A 2 by 2 matrix has four entries. If each … Continue reading →

Review exercise 1.6. (a) For each of the each of the 2 by 2 matrices containing only 0 or 1 as entries, determine whether the matrix is invertible or not. (b) Of the 10 by 10 matrices containing only 0 … Continue reading →

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 … Continue reading →

Review exercise 1.4. Solve the following systems of equations using elimination and back substitution:    and    Answer: We start with the system by subtracting the first equation from the second and third equations to obtain the following system: We … Continue reading →

Review exercise 1.3. Find a 2 by 2 matrix for which and (a) , (b) , and (c) . Answer: (a) We have and . Since we see that is its own inverse. Using the formula for the inverse of … Continue reading →

Review exercise 1.2. Given matrices and as follows find the products and , the inverses , , and . Answer: We have NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear … Continue reading →

Review exercise 1.1. (a) Show the 3 by 3 matrices and for which and . (b) Find the products , , and of the above matrices. Answer: (a) We have and (b) We have (As a point of interest, note … Continue reading →

Exercise 1.7.10. When partial pivoting is used show that for all multipliers in . In addition, show that if for all and then after producing zeros in the first column we have , and in general we have after producing … Continue reading →

Exercise 1.7.9. Given the matrix compare the pivots in standard elimination with those used in partial pivoting. Answer: In the standard elimination process we would use a multiplier (1/.001) to multiply the first row and subtract it from the second: … Continue reading →