Monthly Archives: June 2010

Exercise 1.4.11. State whether the following statements are true or false. If a given statement is false, provide a counterexample to the statement. For two matrices A and B, if the first column of B is identical to the third … Continue reading →

Exercise 1.4.10. Given a matrix A with entries , what are the following entries? first pivot the multiplier that is used to multiply the first row and subtract it from the ith row the value that replaces once the above … Continue reading →

Exercise 1.4.9. Given the following two examples of FORTRAN code DO 10 I=1,N DO 10 J=1,N 10 B(I) = B(I) + A(I,J)*X(J) and DO 10 J=1,N DO 10 I=1,N 10 B(I) = B(I) + A(I,J)*X(J) Do they multiply Ax by … Continue reading →

Exercise 1.4.8. Give non-zero examples of 3×3 matrices with the following properties: diagonal matrix ( if ); symmetric matrix ( for all ); upper triangular matrix ( if ); skew-symmetric matrix ( for all i and j). Answer: Diagonal matrix: … Continue reading →

Exercise 1.4.7. Given the n-dimensional row vector y and column vector x, express their inner product in summation notation. Answer: We have NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear … Continue reading →

Exercise 1.4.6. Find the 3×2 matrix A for which . Find the 3×2 matrix B for which . Answer: We have We also have NOTE: This continues a series of posts containing worked out exercises from the (out of print) … Continue reading →

Exercise 1.4.5. Multiply the following matrices: Considering the matrix A as a system of equations, find a solution to the system Ax = 0, for which the right-hand side of all three equations is zero. Specify whether there is just … Continue reading →

Exercise 1.4.4. Compute the number of multiplications required to multiply an mxn matrix A by an n-dimensional vector x. Also compute the number of multiplications required to multiply A by an nxp matrix B. Answer: The result of multiplying A … Continue reading →

Exercise 1.4.3. Multiply the matrices below: Answer: The first example multiplies a 1×3 matrix by a 3×1 matrix producing a 1×1 matrix, i.e., a scalar value: As noted in the book, the result is equal to the square of the … Continue reading →

Exercise 1.4.2. Multiply the matrices below: Work by columns instead of by rows. Answer: The first example multiplies a 3×2 matrix by a 2×1 matrix (column vector), producing a 3×1 matrix (column vector). Working by rows gives us the following: … Continue reading →