Two matrices are equal if and only if:
The order of the matrices are the same
The corresponding elements of the matrices are the same
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.