Two matrices can be equal, if they have the same order and corresponding elements are equal.
Example :
The above two matrices have the same order 3x2. That is, 3 rows and 2 columns.
Further,
a = u c = w e = y |
b = v d = x f = z |
Problem 1 :
Find the values of w, x, y and z from the following equation.
Solution :
Since the given two matrices are equal, the corresponding elements must be equal.
Then, we have
w = 1 y = 0 |
x = -3 z = 7 |
Problem 2 :
Find the values of w, x, y and z from the following equation.
Solution :
2p = 10 ----(1) 3t = 9 ----(2) |
5p + q = 17 ----(3) 5t + r = 15 ----(4) |
Solve (1) for p.
2p = 10
p = 5
Substitute p = 5 into (3).
5(5) + q =17
25 + q = 17
q = -8
Solve (2) for t.
3t = 9
t = 3
Substitute r = 5 into (3).
5(3) + r =15
15 + r = 15
r = 0
Therefore,
p = 5, q = -8, t = 3, r = 0
Problems 3-5 : Find the values of x, y and z from the following equations.
Problem 3 :
Solution :
(x, y, z) = (3, 12, 3)
Problem 4 :
Solution :
x + y = 6 ----(1)
5 + z = 5 ----(2)
xy = 8 ----(3)
Solve (1) for y.
y = 6 - x ----(4)
Substitute y = 6 - x into (3).
x(6 - x) = 8
6x - x^{2} = 8
x^{2 }- 6x + 8 = 0
x^{2 }- 2x - 4x + 8 = 0
x(x^{ }- 2) - 4(x - 2) = 0
(x - 2)(x - 4) = 0
x - 2 = 0 or x - 4 = 0
x = 2 or x = 4
Substitute x = 2 and 4 into (4).
y = 6 - 2 y = 4 |
y = 6 - 4 y = 2 |
Solve (2) for z.
5 + z = 5
z = 0
Therefore,
(x, y, z) = (2, 4, 0) or (4, 2, 0)
Problem 5 :
Solution :
x + y + z = 9 ----(1)
x + z = 5 ----(2)
y + z = 7 ----(3)
Substitute y + z = 7 into (1)
x + 7 = 9
x = 2
Substitute x = 2 into (2).
2 + z = 5
z = 3
Substitute z = 3 into (3).
y + 3 = 7
y = 4
Therefore,
(x, y, z) = (2, 4, 3)
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