Matrices • Topic 2 of 3

Compatible Matrices

What are compatible matrices? Two matrices are said to be compatible for a given operation if their orders satisfy the conditions required for that operation. Compatibility depends on what operation we want to perform.

For addition and subtraction: Matrices must have the same order. You can only add or subtract matrices when they have exactly the same number of rows AND the same number of columns.

For multiplication (for your information — not in Grade 10 syllabus typically): The number of columns in the first matrix must equal the number of rows in the second matrix.

Real-life analogy: You can only add two tables if they have the same number of rows and columns (same layout). You can't add a 2×3 table to a 3×2 table because the positions don't match!

Examples of compatible matrices for addition:

Matrix AMatrix BCompatible?Why?
2 × 32 × 3✓ YesSame order
3 × 23 × 2✓ YesSame order
2 × 33 × 2✗ NoDifferent orders
1 × 41 × 4✓ YesSame order
2 × 22 × 3✗ NoColumns don't match
For addition the two matrices must have the same order; a 2 by 3 plus 2 by 3 is allowed but 2 by 3 plus 3 by 2 is not Same order → can add or subtract Compatible a b c d + e f g h 2×2 and 2×2 → same order Not compatible a b c d + p q r 2×2 and 3×1 → orders differ vs Rule: A ± B is defined only when A and B have the SAME order. (For a product AB: columns of A must equal rows of B.) Matrix equality: two matrices are equal when they have the same order and every corresponding entry is equal Equality: same order AND matching entries x+y 2 5 x−y = 7 2 5 3 x + y = 7 x − y = 3 Equate corresponding entries → solve: x = 5, y = 2 One matrix equation becomes a set of scalar equations.
Solved Examples
1
Worked Example
Check whether the following pairs of matrices are compatible for addition: (i) A = 2×3, B = 2×3 (ii) P = 3×2, Q = 2×3 (iii) X = 1×4, Y = 1×4
Solution
  1. Step 1: For addition, matrices must have the same order
  2. Step 2: (i) A is 2×3, B is 2×3 → same order → compatible
  3. Step 3: (ii) P is 3×2, Q is 2×3 → different orders → not compatible
  4. Step 4: (iii) X is 1×4, Y is 1×4 → same order → compatible

Answer: (i) Yes, (ii) No, (iii) Yes

2
Worked Example
Can we add a 3×4 matrix and a 3×4 matrix? What about a 3×4 and a 4×3?
Solution
  1. Step 1: First pair: 3×4 and 3×4 → same order (both 3 rows, 4 columns)
  2. Step 2: Therefore, they ARE compatible for addition
  3. Step 3: Second pair: 3×4 and 4×3 → different orders
  4. Step 4: 3×4 has 3 rows, 4×3 has 4 rows → not compatible

Answer: 3×4 and 3×4: Yes; 3×4 and 4×3: No

3
Worked Example
If matrix A is of order 2×3 and A + B is defined, what can you say about the order of B?
Solution
  1. Step 1: For A + B to be defined, A and B must have the same order
  2. Step 2: A has order 2×3 (2 rows, 3 columns)
  3. Step 3: Therefore, B must also have order 2×3

Answer: B must be of order 2×3

Key Points

  • For addition/subtraction, matrices must have the same order
  • Compatible means the operation is possible (defined)
  • If orders are different, addition/subtraction is NOT possible
  • A + B = B + A (commutative property holds for addition)
  • (A + B) + C = A + (B + C) (associative property holds)
  • Zero matrix (same order) is the additive identity
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.Two matrices can be added if they have the:
Explanation: Same order for addition.
Q2.For the product $AB$ to exist:
Explanation: Inner dimensions match.
Q3.A $2\times3$ matrix and a $3\times2$ matrix can be:
Explanation: Multiplication is defined.
Q4.Equal matrices have the same order and:
Explanation: Entry-by-entry equal.
Practice more MCQs → 📝 Assignment · 35 marks