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Current Location > Math Formulas > 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 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.