Equality of Matrices

Simple explanation of equality of matrices with definitions and examples showing when two matrices are considered exactly equal based on order and corresponding elements.

1. What Does It Mean for Two Matrices to Be Equal?

Two matrices are said to be equal when they look exactly the same in every way. That means:

  • They must have the same order (same number of rows and columns).
  • Every corresponding entry must be equal.

If even one entry is different, or the orders do not match, the matrices are not equal.

This idea follows naturally from how matrices are arranged — if two tables of numbers match cell-by-cell, we say they are equal.

2. Formal Definition

If \( A = [a_{ij}] \) and \( B = [b_{ij}] \) are two matrices of order \( m \times n \), then:

\( A = B \iff a_{ij} = b_{ij} \; \text{for all} \; i, j \)

This means that for every position (row i, column j), the entries must be identical.

3. Checking Equality Step by Step

To decide whether two matrices are equal, follow these two steps:

  1. Compare their orders. If the orders differ, stop — they are not equal.
  2. If the orders match, compare each corresponding entry one by one.

If all entries match, the matrices are equal.

3.1. Example 1: Matrices That Are Equal

Consider the matrices:

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

Both matrices are:

  • of order \( 2 \times 2 \)
  • matching element-by-element

So, \( A = B \).

3.2. Example 2: Same Order but Not Equal

Now look at:

\( C = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad D = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix} \)

The orders are the same, but:

  • \( c_{22} = 4 \)
  • \( d_{22} = 5 \)

This single mismatch means \( C \neq D \).

3.3. Example 3: Different Orders

Consider:

\( E = \begin{bmatrix} 1 & 0 & 2 \end{bmatrix}, \quad F = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} \)

Here:

  • \( E \) is of order \( 1 \times 3 \)
  • \( F \) is of order \( 3 \times 1 \)

Even though the numbers look similar, the orders differ, so \( E \neq F \).

4. Why Equality of Matrices Is Important

This concept appears often when:

  • solving matrix equations like \( AX = B \)
  • comparing solutions of linear systems
  • checking steps in transformations and simplifications

Understanding when two matrices are equal helps in verifying results and ensuring logical consistency in calculations.