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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 At.
·  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 B21 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
B21 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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