Matrix Operations & Dot Products

Matrix multiplication is a way of combining two matrices by pairing the rows of the first with the columns of the second — and in NumPy it is a completely different operation from element-wise multiplication.

Learn Matrix Operations & Dot Products in our free NumPy course — a beginner-friendly interactive lesson with worked examples, a practice exercise and a…

Part of the free Numpy course at LearnCodingFast — hands-on lessons with examples you run in your browser, plus practice exercises and a quick quiz.

In this lesson you'll learn why * is element-wise while @ , np.dot , and np.matmul are the true matrix product, plus matrix-vector products and 1D inner products.

This is the single most important distinction in this lesson. The * operator multiplies matching positions together — that's element-wise multiplication. The @ operator performs matrix multiplication , combining whole rows and columns. They almost always give different answers.

Expected output: element-wise [[5 12] [21 32]] , then the matrix product [[19 22] [43 50]] printed three identical times (top-left is 1*5 + 2*7 = 19 ). Shape rule: an (m, n) times an (n, p) matrix gives an (m, p) result.

Multiplying a matrix by a vector transforms that vector. An (m, n) matrix times an (n,) vector produces an (m,) vector. Each entry of the result is the dot product of a row with the vector.

Expected output: [10 10 15] — each value is one row of M dotted with v .

With two 1D arrays np.dot returns a single number , the inner product: multiply matching elements and add them up (great for weighted totals like price × quantity). The identity matrix ( np.eye(n) ) is the matrix version of 1, and the transpose ( A.T ) flips rows and columns to make shapes line up.

Expected output: 32 twice, then the total cost 26.0 ( 2.5*4 + 3.0*2 + 1.0*10 ).

Expected output: A @ I equals A (as floats), the transpose [[1 3] [2 4]] , and A @ A.T = [[5 11] [11 25]] .

Replace each ___ so the program computes the matrix product of A and B.

Expected output: [[2 8] [7 19]] . (Answer: the @ operator.)

❌ ValueError: matmul: ... shapes ... not aligned

The inner dimensions don't match for matrix multiplication.

✅ Fix: an (m, n) matrix needs an (n, p) partner — transpose one operand with .T if needed.

* multiplied positions element-wise instead of doing a matrix product.

✅ Fix: use @ (or np.dot / np.matmul ) for matrix multiplication.

Each row of sales is one store's units sold for three products. Multiply by the price vector to get each store's total revenue with a single matrix-vector product.

Expected output: [ 60 85 150] and Best store index: 2 . (Store 1 is 0*10 + 4*20 + 1*5 = 85 ; store 2 is 5*10 + 5*20 + 0*5 = 150 .)

Lesson 16 complete — you've mastered the matrix product!

You now know that * is element-wise while @ , np.dot , and np.matmul are the true matrix product, and you used matrix-vector and 1D inner products.

🚀 Up next: Conditions — choose, mask, and bound values with np.where , np.select , and np.clip .

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