JENECONKMaths Universe

Matrix Multiplication

Ages 14–15

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Matrix Multiplication

For AB to exist, the number of columns in A must equal the number of rows in B. If A is m × n and B is n × p, then AB is m × p. Each answer entry is a row of A dotted with a column of B: multiply matching values, then add the products. This is different from entry-by-entry multiplication, and matrix multiplication is generally not commutative, so AB may exist while BA does not.

Your goals

  • ✓You can multiply matrices using row-by-column products and check the order of the result.
  • ✓You can use inner dimensions, dot product, outer dimensions correctly.
  • ✓You can write (m × n)(n × p) = m × p before calculating any entries.

Words to know

inner dimensions:
The column count of the first matrix and row count of the second.
dot product:
Multiply matching values in a row and column, then add.
outer dimensions:
The row count of the first and column count of the second.

Examples

  • 11234 5678 = 1×5 + 2×71×6 + 2×83×5 + 4×73×6 + 4×8 = 19224350.
  • 2A (2 × 3) multiplied by B (3 × 1) produces a (2 × 1) matrix because the inner 3s match and the outer 2 and 1 remain.
  • 3A row matrix 241 times a column matrix 352 gives 2×3 + 4×5 + 1×2 = 28.
  • 4If A is 2 × 3 and B is 2 × 2, AB is undefined because 3 does not equal 2. Check the inner dimensions before calculating.
  • 5The identity matrix leaves a compatible matrix unchanged: AI = IA = A, where the dimensions allow both products.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Dimension check

Best for: Deciding whether AB exists
  1. 1.Write the two orders side by side.
  2. 2.Compare the first matrix's columns with the second's rows.
  3. 3.Keep the outside dimensions for the result.

Example: (2 × 3)(3 × 4) is defined and produces a 2 × 4 matrix.

Method 2

Row-by-column dot product

Best for: Calculating each entry accurately
  1. 1.Choose one row from the first matrix.
  2. 2.Choose one column from the second.
  3. 3.Multiply matching values and add the products.

Example: The top-left entry of [[1, 2], [3, 4]][[5, 6], [7, 8]] is 1×5 + 2×7 = 19.

Check these

  • • Do not multiply entries in the same position or assume AB = BA.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own matrix multiplication example, solve it, and explain how you checked it.

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