Find the determinant of the matrices below by inspection. Give your reason in each case.

Suppose matrix product AB is defined. (a) If A is 5×6 and B is a column matrix, give the dimensions of B and AB. (b) If A is the identity matrix and B is 5×5, what size is A? (c) If A is 7×6 and AB is 7×5, what size is B?
Question 2.
Find the determinant of the matrices below by inspection. Give your reason in each case.
1
B =   
1 5 3 −2 3 11 −4 13 17 −5 2 −23 −6 −22 8 −26
    C =
 
6 0 0 −1 8 0 2 10 1
  D = 
3 0 0 0 7 0 0 0 −2
 .
Question 3.
Determine if D is invertible and find its inverse if possible D =  1 3 0 0 1 1 2 0 0  .
Question 4. Find the derivative of the following function: (x2)sinx.
Question 5. (a) Find the indefinite integral of f(x) = 11x + 6x2 −√x−e2x. (b) Hence calculateR1 0 (11x + 6x2 −√x−e2x)dx. Question 6.
Evaluate the following integrals using the given substitutions. (a) Z (3x2 + 10x)dx x3 + 5x2 + 18 , substitution u = x3 + 5x2 + 18; (b) Z(14x + 4)cos(7x2 + 4x)dx, substitution u = 7x2 + 4x.
2
Full Answer Questions [60 marks] • Hand write full solutions neatly using careful language and correct notation. Do not type-set unless your software offers quality notation. • We expect a high standard of communication. Look at the tutorial worked examples in the study desk to see the level you should aim at. Up to 15% of the marks may be deducted for poor language and notation.
Algebra
[30 marks]
Question 7.
Solve the following system of equations by hand. Use the Gaussian elimination, on the augmented matrix, and write the row operation you used next to each new row.
x


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