Understanding Engineering Mathematics

2. For the matrices

A=

[−10 1

21 0

32 − 1

]

B=

[ 423

− 241

321

]

evaluate

(i) A+B (ii) A−B (iii) 3A+ 2 B

(iv) AB (v) BA (vi) A^2

Answers

1.x=^32 ,y=^12 ,X=2,Y= 1

2. (i)

[

324

051

640

]

(ii)

[

− 5 − 2 − 2

4 − 3 − 1

00 − 2

]

(iii)

[

54 9

211 2

15 10 − 1

]

(iv)

[− 10 − 2

68 7

512 10

]

(v)

[ 98 1

13 6 − 3

44 2

]

(vi)

[ 42 − 2

01 2

−20 4

]

13.4 Determinants

First, an admission. The full theory of determinants is conceptually quite straightforward,
but it is very intricate. I will not, therefore, be going into much depth, and I will ask you
to make a leap of faith on occasions. However, determinants have great theoretical impor-
tance – in defining the inverse matrix, and in eigenvalue theory. It may appear therefore
that I am selling you short in skimming over determinants. However, inpracticeno one
uses determinants to invert matrices or find eigenvalues for real problems – there are far
more effective means studied at more advanced level. Another point is that the definition
of a determinant appears a little complicated and so I’ll work up to it gently, so that you
can see where it comes from. To help you interpret the symbols, a numerical example is
done in parallel.

Consider the simple system of
equations:

a 11 x 1 +a 12 x 2 =b 1 (i)

a 21 x 1 +a 22 x 2 =b 2 (ii)

Multiply (i) by a 21 , (ii) by a 11 and
subtract

(a 11 a 22 −a 21 a 12 )x 2 =a 11 b 2 −a 21 b 1

Problem 13.8

Solve the pair of equations

3 x 1 Y 2 x 2 = 1 .i/

x 1 − 2 x 2 = 2 .ii/

Multiply (i) by 1, (ii) by 3 and

subtract

( 3 ×(− 2 )− 1 × 2 )x 2 = 3 × 2 − 1 × 1