Multiply a Row by a Column

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Multiply [[1,2],[0,1]] by column [2,3]. What is the first component? Candidate answers: (A) 16 (B) 7 (C) 8 (D) 9.
• Independent check inputs: Multiply [[1,2],[0,1]] by column [3,4]. What is the first component? Candidate answers: (A) 10 (B) 11 (C) 12 (D) 22.

Learn the idea

Matrices organize linear operations and systems through rows and columns. Multiplication uses row-column products, a determinant tests two-dimensional invertibility, and valid row operations preserve a system’s solution set when applied to every entry. A matrix acts on an input column by taking each row’s dot product with that column. The resulting output has one component for each row of the matrix.

Here is the method on the supplied teaching case. The first output is the first row dot the column: 1(2)+2(3)=8. The second output is 3; entries are not multiplied only by position.

Row and column dimensions must be compatible. Multiplying matching positions without summing does not give a matrix product; the order of matrix multiplication generally matters.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Multiply [[1,2],[0,1]] by column [9,10]. What is the first component? Candidate answers: (A) 28 (B) 29 (C) 30 (D) 58. Show the mathematical reason, not only the selected answer. Expected result: 29. The first output is the first row dot the column: 1(9)+2(10)=29. The second output is 10; entries are not multiplied only by position.

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