232 Higher Engineering Mathematics
(a) Adding the corresponding elements gives:
2 −1
−7
4
+
−3
0
7 −4
=
2 + (−3) −1 + 0
−7 + 7
4+ (−4)
=
−1 −1
0
0
(b) Adding the corresponding elements gives:
⎛
⎝
3 1 −4
4 3
1
1 4 −3
⎞
⎠ +
⎛
⎝
2 7 −5
−2 1
0
6 3
4
⎞
⎠
=
⎛
⎝
3 + 2
1+ 7 −4 + (−5)
4 + (−2) 3 + 1
1+ 0
1 + 6
4+ 3 −3 + 4
⎞
⎠
=
⎛
⎝
5 8 −9
2 4
1
7 7
1
⎞
⎠
(ii) Subtraction of matrices
If A is a matrix and B is another matrix, then (A − B)
is a single matrix formed by subtracting the elements of
B from the corresponding elements of A.
Problem 2. Subtract
(a)
−3
0
7 −4
from
2 −1
−7
4
and
(b)
⎛
⎝
2 7 −5
−2 1
0
6 3
4
⎞
⎠ from
⎛
⎝
3 1 −4
4 3
1
1 4 −3
⎞
⎠
To find matrix A minus matrix B, the elements of B are
taken from the corresponding elements of A. Thus:
(a)
2 −1
−7
4
−
−3
0
7 −4
=
2 − (−3) −1 − 0
−7 − 7
4− (−4)
=
5 −1
−14
8
(b)
⎛
⎝
3 1 −4
4 3
1
1 4 −3
⎞
⎠ −
⎛
⎝
2 7 −5
−2 1
0
6 3
4
⎞
⎠
=
⎛
⎝
3 − 2
1− 7 −4 − (−5)
4 − (−2) 3 − 1
1− 0
1 − 6
4− 3 −3 − 4
⎞
⎠
=
⎛
⎝
1 −6
1
6
2
1
−5
1 −7
⎞
⎠
Problem 3. If
A =
−3
0
7 −4
, B =
2 −1
−7
4
and
C =
1
0
−2 −4
find A + B − C.
A + B =
−1 −1
0
0
(from Problem 1)
Hence, A + B − C =
−1 −1
0
0
−
1
0
−2 −4
=
−1 − 1
−1 − 0
0 − (−2)
0 − (−4)
=
−2 −1
2
4
Alternatively A + B − C
=
−3
0
7 −4
+
2 −1
−7
4
−
1
0
−2 −4
=
−3 + 2 − 1
0+ (−1) − 0
7 + (−7) − (−2) −4 + 4 − (−4)
=
−2 −1
2
4
as obtained previously
(iii) Multiplication
When a matrix is multiplied by a number, called scalar
multiplication, a single matrix results in which each
element of the original matrix has been multiplied by
the number.
Problem 4. If A =
−3
0
7 −4
,
B =
2 −1
−7
4
and C =
⎛
⎝
1
0
−2 −4
⎞
⎠ find
2 A − 3B + 4C.