Matrices and Operations

Explore matrix dimensions, addition and scalar multiplication, matrix multiplication.

Lesson 2 of 53 parts Practice Flashcards

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Matrix Dimensions

Objective: Explain matrix dimensions and use it to solve or interpret a problem.

An m×n matrix has m rows and n columns. Addition requires matching dimensions and multiplication requires matching inner dimensions. m×p.

Worked examples

Try itMatrices and Operations — practice 1.1: A 2×6 matrix multiplies a 6×5 matrix. How many columns in the product?Show answer

Step 1: The inner dimensions 6 match. Step 2: The product has shape 2×5. Step 3: It has 5 columns.

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Matrix Dimensions

Objective: Explain matrix dimensions and use it to solve or interpret a problem.

An m×n matrix has m rows and n columns. Addition requires matching dimensions and multiplication requires matching inner dimensions. m×p.

Worked examples

Try itMatrices and Operations — practice 1.1: A 2×6 matrix multiplies a 6×5 matrix. How many columns in the product?Show answer

Step 1: The inner dimensions 6 match. Step 2: The product has shape 2×5. Step 3: It has 5 columns.

Addition and Scalar Multiplication

Objective: Explain addition and scalar multiplication and use it to solve or interpret a problem.

Add corresponding entries of equally shaped matrices; a scalar multiplies every entry. These operations preserve matrix dimensions. No, their shapes differ.

Worked examples

Try itMatrices and Operations — practice 2.1: Find top-left entry of [[3,7],[1,2]]+[[8,1],[3,4]].Show answer

Step 1: Add corresponding top-left entries 3+8. Step 2: The resulting top-left entry is 11. Step 3: Matrix dimensions match: both are 2×2.

Matrix Multiplication

Objective: Explain matrix multiplication and use it to solve or interpret a problem.

Product entry (i,j) is a dot product of row i of the left matrix and column j of the right. Matrix multiplication need not commute. No, order can change or make a product undefined.

Worked examples

Try itMatrices and Operations — practice 3.1: Find top-left entry of [[4,3],[1,2]]·[[11,1],[2,3]].Show answer

Step 1: Take row 1 of the left and column 1 of the right. Step 2: Compute 4·11+3·2. Step 3: The result is 50.

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