Lesson 9: Multiplying Matrices

New Concept

What’s next

Next, you’ll use this process to solve problems, explore matrix properties, and apply multiplication to a real-world shopping scenario.

A matrix $A$ can be multiplied by a matrix $B$ if the number of columns in $A$ equals the number of rows in $B$. To find an element in the product matrix, multiply the elements of a row from the first matrix by the corresponding elements of a column from the second matrix, then add the products.

Find $AB$ if $A = \begin{bmatrix} 1 & -2 \\ 4 & 0 \\ 2 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 1 \\ 2 & -4 \end{bmatrix}$. The product is $AB = \begin{bmatrix} 1(5)+(-2)(2) & 1(1)+(-2)(-4) \\ 4(5)+0(2) & 4(1)+0(-4) \\ 2(5)+3(2) & 2(1)+3(-4) \end{bmatrix} = \begin{bmatrix} 1 & 9 \\ 20 & 4 \\ 16 & -10 \end{bmatrix}$.
Find $XY$ if $X = \begin{bmatrix} 2 & -1 \\ 3 & 5 \end{bmatrix}$ and $Y = \begin{bmatrix} 0 & 4 \\ 1 & -2 \end{bmatrix}$. The product is $XY = \begin{bmatrix} 2(0)+(-1)(1) & 2(4)+(-1)(-2) \\ 3(0)+5(1) & 3(4)+5(-2) \end{bmatrix} = \begin{bmatrix} -1 & 10 \\ 5 & 2 \end{bmatrix}$.

Think of it as a 'row-hugs-column' dance! Each number in a row from the first matrix finds a partner in the corresponding column of the second matrix. They multiply, and all the pairs in that dance sum up to create one number in the final matrix. This repeats for every row and every column until the new matrix is complete.

When two matrices can be multiplied, the resulting matrix is called the product matrix. If an $m \times n$ matrix is multiplied by an $n \times p$ matrix, the dimensions of the product matrix are determined by the 'outside' numbers, resulting in an $m \times p$ matrix.

Matrix $A$ is $2 \times 5$ and Matrix $B$ is $5 \times 4$. The product $AB$ is defined and the product matrix will have dimensions $2 \times 4$.
Matrix $C$ is $3 \times 1$ and Matrix $D$ is $2 \times 3$. The product $CD$ is not defined because the inner dimensions (1 and 2) do not match.
Matrix $X$ is $1 \times 7$ and Matrix $Y$ is $7 \times 1$. The product $XY$ is defined and the product matrix will have dimensions $1 \times 1$.

Before you do any math, play matchmaker with the dimensions! If you have a $4 \times 3$ matrix and a $3 \times 5$ matrix, the inner '3s' match, so multiplication is possible. The outer numbers, 4 and 5, give you a sneak peek at the answer's size: your final product matrix will be a glorious $4 \times 5$ matrix.

The multiplicative identity matrix, denoted as $I$, is a square matrix where all elements on the main diagonal are ones and all other elements are zeros. The product of any matrix $A$ and the appropriate identity matrix $I$ is always $A$.

For $A = \begin{bmatrix} 5 & 1 \\ -2 & 8 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, then $AI = \begin{bmatrix} 5 & 1 \\ -2 & 8 \end{bmatrix}\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 5 & 1 \\ -2 & 8 \end{bmatrix}$.
For $B = \begin{bmatrix} 4 & -1 & 7 \\ 2 & 0 & -5 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$, then $BI = \begin{bmatrix} 4 & -1 & 7 \\ 2 & 0 & -5 \end{bmatrix}$.

Meet the superhero of matrices: the Identity Matrix, or $I$! It's the matrix version of the number 1. Multiplying any matrix by $I$ is like giving it a high-five—it leaves the original matrix completely unchanged. This special square matrix has a slick diagonal of ones and is your go-to for preserving a matrix's identity during calculations.

Matrix multiplication is a powerful tool for organizing and solving real-world problems involving multiple quantities and costs. By setting up a quantity matrix and a cost matrix, their product can efficiently calculate total expenses for different categories or individuals in a single, structured operation.

Three friends buy school supplies. Ann buys 2 pens, 1 notebook. Ben buys 1 pen, 3 notebooks. Pens are 2 dollars, notebooks are 4 dollars. $\begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} 2 \\ 4 \end{bmatrix} = \begin{bmatrix} 2(2)+1(4) \\ 1(2)+3(4) \end{bmatrix} = \begin{bmatrix} 8 \\ 14 \end{bmatrix}$. Ann spends 8 dollars, Ben spends 14 dollars.
Uniforms cost 30 dollars for jerseys and 20 dollars for shorts. Team A needs 10 of each. Team B needs 12 of each. $\begin{bmatrix} 10 & 10 \\ 12 & 12 \end{bmatrix}\begin{bmatrix} 30 \\ 20 \end{bmatrix} = \begin{bmatrix} 500 \\ 600 \end{bmatrix}$. Team A spends 500 dollars, Team B spends 600 dollars.

Who knew matrices could manage your shopping trip? Imagine you have a list of items each friend wants (the quantity matrix) and a list of prices (the cost matrix). Multiplying them together instantly calculates the total bill for each person! It's like having a super-organized personal shopper that handles all the money math in one clean step.