1.	The matrices A, B, X are given by
	A =  B =  X =  where a, b, c, d ∈ _.
	Given that AX + X = Β, find the exact values of a, b, c and d.
(Total 8 marks)

2.	Let A = .
 
(a)	Find the value of x for which A^–1 does not exist.
(3)
 
(b)	Given that A = A^–1, find x.
(5)
(Total 8 marks)

3.	Consider the function f(x) = px^3 + qx^2 + rx. Part of the graph of f is shown below.
		
	The graph passes through the origin O and the points A(–2, –8), B(1, –2) and C(2, 0).
 
(a)	Find three linear equations in p, q and r.
(4)
 
(b)	Hence find the value of p, of q and of r.
(3)
(Total 7 marks)

4.	Let f(x) = ax^2 + bx + c where a, b and c are rational numbers.
(a)	The point P(–4, 3) lies on the curve of f. Show that 16a –4b + c = 3.
(2)
 
(b)	The points Q(6, 3) and R(–2, –1) also lie on the curve of f. Write down two other linear equations in a, b and c.
(2)
 
(c)	These three equations may be written as a matrix equation in the form AX = B,
where X = .
(i)	Write down the matrices A and B.
(ii)	Write down A^–1.
(iii)	Hence or otherwise, find f(x).
(8)
 
(d)	Write f(x) in the form f(x) = a(x – h)^2 + k, where a, h and k are rational numbers.
(3)
(Total 15 marks)

5.	Let A = and B = .
(a)	Find AB.
(b)	The matrix C =  and 2AB = C. Find the value of x.
(Total 6 marks)

6.	Let A = .
(a)	Write down A^–1.
(2)
 
	The matrix B satisfies the equation  = A, where I is the 3 × 3 identity matrix.
(b)	(i)	Show that B = –2(A^–1 – I).
(ii)	Find B.
(iii)	Write down det B.
(iv)	Hence, explain why B^–1 exists.
(6)
 
	Let BX = C, where X =  and C = .
(c)	(i)	Find X.
(ii)	Write down a system of equations whose solution is represented by X.
(5)
(Total 13 marks)

7.	Let A = .
(a)	Find
(i)	A^−1;
(ii)	A^2.
(4)
 
Let B = .
(b)	Given that 2A + B = , find the value of p and of q.
(3)
 
(c)	Hence find A^−1B.
(2)
 
(d)	Let X be a 2 × 2 matrix such that AX = B. Find X.
(2)
(Total 11 marks)

IB Questionbank Maths SL