Equality of Two Matrices

In mathematics, particularly in linear algebra, the concept of equality of two matrices is fundamental. Two matrices are considered equal if they have the same dimensions and their corresponding elements are equal.

Definition

Two matrices $A$ and $B$ are equal, denoted as $A = B$, if and only if:

1. They have the same number of rows and columns (i.e., the same order).
2. Corresponding elements are equal, which means $a_{ij} = b_{ij}$ for all $i$ and $j$, where $a_{ij}$ is the element in the $i$-th row and $j$-th column of matrix $A$, and $b_{ij}$ is the element in the $i$-th row and $j$-th column of matrix $B$.

Mathematically, if $A$ is an $m \times n$ matrix and $B$ is a $p \times q$ matrix, then $A = B$ if and only if $m = p$, $n = q$, and $a_{ij} = b_{ij}$ for all $1 \leq i \leq m$ and $1 \leq j \leq n$.

Formulas

The equality of two matrices can be represented as:

$$ A = B \iff \begin{cases} a_{ij} = b_{ij} & \text{for all } i, j \ m = p & \text{(number of rows)} \ n = q & \text{(number of columns)} \end{cases} $$

Examples

Let's consider two matrices $A$ and $B$:

$$ A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} $$

Here, $A$ and $B$ are equal because they are of the same order (2x2) and all corresponding elements are equal.

Now, let's consider another example where matrices $C$ and $D$ are not equal:

$$ C = \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix}, \quad D = \begin{bmatrix} 5 & 6 \ 7 & 9 \end{bmatrix} $$

In this case, $C \neq D$ because the element in the second row and second column is different ($c_{22} = 8$ while $d_{22} = 9$).

Table of Differences and Important Points

Aspect	Equality Condition
Order of Matrices	The number of rows and columns must be the same for both matrices.
Corresponding Elements	Each element in one matrix must be equal to the corresponding element in the other matrix.
Notation	Equal matrices are denoted by the symbol $=$, while non-equal matrices are denoted by $\neq$.
Transitivity	If $A = B$ and $B = C$, then $A = C$.
Reflexivity	A matrix is always equal to itself, i.e., $A = A$.
Symmetry	If $A = B$, then $B = A$.

Conclusion

Equality of matrices is a straightforward concept, but it is crucial for many operations in linear algebra. When working with matrices, always ensure that you are comparing matrices of the same order and that their corresponding elements match. This concept is not only essential for theoretical mathematics but also for practical applications in various fields such as computer science, physics, and engineering.