# matrices 2

```1.
0 2 
 .
Consider the matrix A = 
 a 1
(a)
Find the matrix A2.
(2)
(b)
If det A2 = 16, determine the possible values of a.
(3)
(Total 5 marks)
2.
Consider the matrices
3  2
1 3 
, B  
 .
A  
5  4
 2  2
(a)
Find BA.
(2)
(b)
Calculate det (BA).
(2)
(c)
Find A(A–1B + 2A–1)A.
(3)
(Total 7 marks)
3.
Let A, B and C be non-singular 2&times;2 matrices, I the 2&times;2 identity matrix and k a scalar. The
following statements are incorrect. For each statement, write down the correct version of the
right hand side.
(a)
(A + B)2 = A2 + 2AB + B2
(2)
(b)
(A – kI)3 = A3 – 3kA2 + 3k2A – k3
(2)
IB Questionbank Mathematics Higher Level 3rd edition
1
(c)
CA = B  C =
B
A
(2)
(Total 6 marks)
4.
(a)
Write down the inverse of the matrix
1  3 1 


A =  2 2  1
1  5 3 


(2)
(b)
Hence, find the point of intersection of the three planes.
x – 3y + z = 1
2x + 2y – z = 2
x – 5y + 3z = 3
(3)
(c)
A fourth plane with equation x + y + z = d passes through the point of intersection. Find
the value of d.
(1)
(Total 6 marks)
5.
 ex
Consider the matrix A = 
x
2  e
e x 
 , where x 
1 
.
Find the value of x for which A is singular.
(Total 6 marks)
6.
The square matrix X is such that X3 = 0. Show that the inverse of the matrix (I – X) is
I + X + X2.
(Total 6 marks)
7.
Matrices A, B and C are defined as
IB Questionbank Mathematics Higher Level 3rd edition
2
5 1
 1
 1 2  1
 8 




 
A   3  1 3 , B   3  1 0 , C   0  .
  9 3 7
0 3 1 
  4




 
(a)
a 0 0


Given that AB =  0 a 0  , find a.
0 0 a


(1)
(b)
Hence, or otherwise, find A–1.
(2)
(c)
Find the matrix X, such that AX = C.
(2)
(Total 5 marks)
8.
 2  1
 and that M2 – 6M + kI = 0 find k.
Given that M = 

3
4


(Total 5 marks)
IB Questionbank Mathematics Higher Level 3rd edition
3
```