Matrix operations

Objective. Combine compatible matrices and interpret dimensions.

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2. Worked example
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Learn: the key idea

A matrix is a rectangular array of numbers, and operations on matrices are only defined when their dimensions are compatible. Matrices of the same dimensions are added or subtracted entry by entry, while matrix multiplication AB is defined only when the number of columns of A equals the number of rows of B, with entries formed from dot products of rows and columns. For example, a 2×3 matrix can multiply a 3×2 matrix to give a 2×2 result. Matrix multiplication is generally not commutative, so AB ≠ BA in general, which must always be checked rather than assumed.

Worked example

Given matrices A = [[2,1],[0,3]] and B = [[1,4],[2,−1]], find the matrix product AB.

1 $Top-left entry: (2\times1) + (1\times2) = 2 + 2 = 4$
2 $Top-right entry: (2\times4) + (1\times-1) = 8 - 1 = 7$
3 $Bottom-left entry: (0\times1) + (3\times2) = 0 + 6 = 6$
4 $Bottom-right entry: (0\times4) + (3\times-1) = 0 - 3 = -3$

Answer: AB = [[4,7],[6,−3]]

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Common mistake. Students multiply matrices entry-by-entry like addition, e.g. computing [[1,2],[3,4]]×[[5,6],[7,8]] as [[5,12],[21,32]] instead of using row-by-column multiplication. Always form each entry as a sum of row-times-column products.

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