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Definition of a Matrix
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Math Formulas
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Linear Algebra
> Equality of Matrices
Equality of Matrices
Two matrices are equal if all three of the following conditions are met:
· Each matrix has the same number of rows.
· Each matrix has the same number of columns.
· Corresponding elements within each matrix are equal.
Consider the three matrices shown below.
If
A = B
then we know that x = 34 and y = 54, since corresponding elements of equal matrices are also equal.
We know that matrix
C
is not equal to
A
or
B
, because
C
has more columns.
Note:
· Two equal matrices are exactly the same.
· If rows are changed into columns and columns into rows, we get a transpose matrix. If the original matrix is A, its transpose is usually denoted by A
^{'}
or A
^{t}
.
· If two matrices are of the same order (no condition on elements) they are said to be comparable.
· If the given matrix A is of the order m x n, then its transpose will be of the order n x m.
Example 1
: The notation below describes two matrices
A
and
B
.
where i= 1, 2, 3 and j = 1, 2
Which of the following statements about
A
and
B
are true?
I. Matrix
A
has 5 elements.
II. The dimension of matrix
B
is 4
×
2.
III. In matrix
B
, element
B
_{21}
is equal to 222.
IV. Matrix
A
and
B
are equal.
(A) I only
(B) II only
(C) III only
(D) All of the above
(E) None of the above
Solution:
The correct answer is (E)
Matrix
A
has 3 rows and 2 columns; that is, 3 rows each with 2 elements. This adds up to 6 elements not 5.
The dimension of matrix
B
is 2
×
4 and not 4
×
2, which means that matrix
B
has 2 rows and 4 columns and not 4 rows and 2 columns.
Element
B
_{21}
refers to the first element in the second row of matrix
B
, which is equal to 555 but not 222.
Matrix
A
and
B
cannot be equal because, we don’t know anything about the entries of matrix
A
. They are just unknown for us. Moreover, their orders are also different.
Example 2
: Find the values of “a” and “b” if [a 3] = [4 b].
Solution: If [a 3] = [4 b], then the corresponding elements of the matrices are equal, thus a = 4 and b = 3.
Example 3
: Determine the values of a, b, c and d, so that the following equation becomes valid.
Solution:
If the two matrices are equal then the corresponding elements are equal too, thus we have:
a = 5
,
a + c = 4
,
b – 2d = 1
and
2b = 6
Insert the value of a into
a + c = 4
will give:
c = –1
And isolating
b
from
2b = 6
will give:
b = 3
Insert
b = 3
into
b–2d = 1
will give:
d = 1
Thus the two given matrices will be equal if a = 5, b = 3, c = –1 and d = 1.
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