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We proceed the same way to obtain the second row of $\text{\hspace{0.17em}}AB.\text{\hspace{0.17em}}$ In other words, row 2 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ times column 1 of $\text{\hspace{0.17em}}B;\text{\hspace{0.17em}}$ row 2 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ times column 2 of $\text{\hspace{0.17em}}B;\text{\hspace{0.17em}}$ row 2 of $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ times column 3 of $\text{\hspace{0.17em}}B.\text{\hspace{0.17em}}$ When complete, the product matrix will be
For the matrices $\text{\hspace{0.17em}}A,B,\text{}$ and $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ the following properties hold.
Note that matrix multiplication is not commutative.
Multiply matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and matrix $\text{\hspace{0.17em}}B.$
First, we check the dimensions of the matrices. Matrix $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ has dimensions $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}2\text{\hspace{0.17em}}$ and matrix $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ has dimensions $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions $\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}2.$
We perform the operations outlined previously.
Given $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B:$
Is it possible for AB to be defined but not BA ?
Yes, consider a matrix A with dimension $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}4\text{\hspace{0.17em}}$ and matrix B with dimension $\text{\hspace{0.17em}}4\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}2.\text{\hspace{0.17em}}$ For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.
Let’s return to the problem presented at the opening of this section. We have [link] , representing the equipment needs of two soccer teams.
Wildcats | Mud Cats | |
---|---|---|
Goals | 6 | 10 |
Balls | 30 | 24 |
Jerseys | 14 | 20 |
We are also given the prices of the equipment, as shown in [link] .
Goal | $300 |
Ball | $10 |
Jersey | $30 |
We will convert the data to matrices. Thus, the equipment need matrix is written as
The cost matrix is written as
We perform matrix multiplication to obtain costs for the equipment.
The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.
Given a matrix operation, evaluate using a calculator.
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