Matrix addition & multiplication

Japanese school year: University year 1

What you learn

This section introduces the definition of matrices along with the rules for matrix addition, scalar multiplication, and matrix multiplication. These operations are fundamental for representing linear transformations and systems of equations compactly. Familiarity with basic arithmetic and the dot product of vectors will facilitate your understanding.

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Key points

This rule shows how to add two matrices $A$ and $B$ . When two matrices have the same size, you simply add the numbers at the matching positions $a_{ij}$ and $b_{ij}$ together.

(A + B)_{ij} = a_{ij} + b_{ij}

This formula calculates the entry in row $i$ and column $j$ of the matrix product $AB$ . To find each number, multiply row $i$ of matrix $A$ by column $j$ of matrix $B$ in order and sum them up.

(AB)_{ij} = \sum_{k=1}^n a_{ik}b_{kj}

This formula shows that parentheses can be expanded in matrix multiplication using the distributive law. While useful for simplifying expressions, the multiplication order cannot be changed, so $AB$ must not be swapped into $BA$ .

A(B + C) = AB + AC

This rule (associative law) shows that when multiplying three matrices, you get the same result whether you compute $AB$ first or $BC$ first. You can start with whichever pair is easier, but the order of matrices $A, B, C$ cannot change.

(AB)C = A(BC)

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Practice 1Coming soon