Matrix Addition and Multiplication
Aligned to N-VM.C.8 — Common Core State Standards for Mathematics.
What this lesson teaches
This is a (+) advanced standard. Add or subtract matrices of the same size entry by entry. Multiply two matrices when the first matrix's number of columns equals the second's number of rows; each entry is a row-times-column dot product.
Worked examples
Example 1
Multiply [[1, 2], [3, 4]] × [[5, 6], [7, 8]]. Top-left = 1×5 + 2×7 = 19, top-right = 1×6 + 2×8 = 22, bottom-left = 3×5 + 4×7 = 43, bottom-right = 3×6 + 4×8 = 50, giving [[19, 22], [43, 50]].
Example 2
Add [[2, 1], [0, 5]] + [[3, −1], [4, 2]] = [[5, 0], [4, 7]] — addition works entry by entry. Now multiply a 2×3 by a 3×1: [[1, 2, 3], [4, 5, 6]] × [[1], [0], [2]] gives [[1 + 0 + 6], [4 + 0 + 12]] = [[7], [16]], a 2×1 result.
Example 3
Common mistake: assuming matrix multiplication commutes. For A = [[1, 2], [0, 1]] and B = [[1, 0], [1, 1]], AB = [[3, 2], [1, 1]] but BA = [[1, 2], [1, 3]] — genuinely different matrices. And multiplication is only DEFINED when the inner dimensions match: a 2×3 times a 3×1 works, while a 3×1 times a 2×3 does not exist at all. Check dimensions before multiplying.
Practice questions
- Add [[1, 0], [2, 3]] + [[4, 5], [1, 1]].
- Can you multiply a 2×3 matrix by a 2×2 matrix? Explain.
- Multiply [[2, 0], [1, 3]] × [[1, 4], [2, 1]].
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Suma y multiplicación de matrices
Este es un estándar avanzado (+). Suma o resta matrices del mismo tamaño entrada por entrada. Multiplica dos matrices cuando el número de columnas de la primera es igual al número de filas de la segunda; cada entrada es un producto punto de fila por columna.
Ejemplo 1: Multiplica [[1, 2], [3, 4]] × [[5, 6], [7, 8]]. Arriba-izquierda = 1×5 + 2×7 = 19, arriba-derecha = 1×6 + 2×8 = 22, abajo-izquierda = 3×5 + 4×7 = 43, abajo-derecha = 3×6 + 4×8 = 50, lo que da [[19, 22], [43, 50]].
Ejemplo 2: Suma [[2, 1], [0, 5]] + [[3, −1], [4, 2]] = [[5, 0], [4, 7]] — la suma va entrada por entrada. Ahora multiplica una 2×3 por una 3×1: [[1, 2, 3], [4, 5, 6]] × [[1], [0], [2]] da [[1 + 0 + 6], [4 + 0 + 12]] = [[7], [16]], un resultado 2×1.
Ejemplo 3: Error común: suponer que la multiplicación de matrices es conmutativa. Para A = [[1, 2], [0, 1]] y B = [[1, 0], [1, 1]], AB = [[3, 2], [1, 1]] pero BA = [[1, 2], [1, 3]] — matrices realmente distintas. Y la multiplicación solo está DEFINIDA cuando las dimensiones interiores coinciden: una 2×3 por una 3×1 funciona, mientras que una 3×1 por una 2×3 ni siquiera existe. Revisa las dimensiones antes de multiplicar.
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