Business Mathematics and Basic Statistics · Ch 10 — Matrices (up to 2×2)

Multiplication of Two Matrices (up to Order 2×2)

Conformability for multiplication

Two matrices $A$ and $B$ can be multiplied, in the order $AB$ , only if the number of COLUMNS of $A$ equals the number of ROWS of $B$ . Two $2\times2$ matrices always satisfy this (2 columns in the first matches 2 rows in the second), so within this chapter's scope, any two $2\times2$ matrices can always be multiplied together.

The row-by-column rule

Each element of the product $AB$ is found by taking a ROW of $A$ and a COLUMN of $B$ , multiplying corresponding entries together, and adding the results:

AB = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}\begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \end{pmatrix}

The element in row $i$ , column $j$ of $AB$ always comes from ROW $i$ of $A$ combined with COLUMN $j$ of $B$ .

Multiplication is NOT commutative

Unlike ordinary numbers, matrix multiplication generally does not commute: $AB$ and $BA$ are usually DIFFERENT matrices, even when both products exist and are of the same order. The order in which two matrices are multiplied always matters.

Note

Order matters

Always read a matrix product left to right exactly as written. $AB$ means "rows of $A$ combined with columns of $B$ " — reversing this to "rows of $B$ combined with columns of $A$ " generally gives a completely different answer, not just the same answer written differently.

The identity matrix under multiplication …

Definition 1Product of Two Matrices (AB)

For 2×2 matrices A and B, each element of AB is found using the row-by-column rule: multiply corresponding entries of a row of A and …

Definition 2Non-Commutativity

In general, AB ≠ BA for matrices — the order of multiplication changes the result, unlike ordinary num …

Start your 14-day free trial to unlock the full solution →

← Previous · 4Scalar Multiplication of a Matrix Next →Q10 — Find the values of a, b, c and d if 2ab-3; c+14d = 85; 620 .