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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

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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