A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns. To multiply a matrix, you must multiply the elements (or numbers) in the first row of the matrix by the elements in the second row of the matrix and add up the product. You can multiply matrices with just a few easy steps that require correct addition, multiplication and placement of the results.
Step
Step 1. Make sure that the matrices are multipliable
You can only multiply a matrix if the number of columns of the first matrix is equal to the number of rows of the second matrix.
These matrices can be multiplied because the first matrix, Matrix A, has 3 columns, while the second matrix, Matrix B, has 3 rows
Step 2. Mark the dimensions of the matrix product
Create a new empty matrix, which will mark the dimensions of the product of the two matrices. The matrix that represents the product of Matrix A and Matrix B will have the same number of rows as the first matrix and the same number of columns as the second matrix. You can draw blank boxes to show the number of rows and columns in this matrix.
- Matrix A has 2 rows, so the result of multiplying the matrix will have 2 rows.
- Matrix B has 2 columns, so the result of multiplying the matrix will have 2 columns.
- The result of the matrix product will have 2 rows and 2 columns.
Step 3. Find the result of the first dot product
To find the result of the first dot product, you must multiply the first element in the first row by the first element in the first column, the second element in the first row by the second element in the first column, and the third element in the first row by the third element in the first column. Then, add up the multiplication results to find dot product (dot).
Suppose, you have decided to first calculate the elements in the second row and second column (bottom right) of the matrix product. Here's how you do it:
- 6 x -5 = -30
- 1 x 0 = 0
- -2 x 2 = -4
- -30 + 0 + (-4) = -34
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The result of the dot product (dot) is -34 and this result is written at the bottom right of the matrix multiplication result.
When you multiply a matrix, the dot product will be written in the row position of the first matrix and the column position of the second matrix. For example, when you know the dot product of the bottom row of Matrix A and the right column of Matrix B, the answer, -34, is written in the bottom row and right column of the matrix product
Step 4. Find the result of the second dot product
Let's say you want to find the term at the bottom left of the matrix product. To find this term, you just need to multiply the elements in the bottom row of the first matrix by the elements in the first column of the second matrix and then add them up. Use the same method as multiplying the first row and column – find again dot product (do t)his.
- 6 x 4 = 24
- 1 x (-3) = -3
- (-2) x 1 = -2
- 24 + (-3) + (-2) = 19
- The result of the dot product is -19 and this result is written at the bottom left of the matrix product.
Step 5. Find the other two dot products
To find the term in the upper left of the matrix product, start by finding the dot product of the top row of Matrix A and the left column of Matrix B. Here's how you do it:
- 2 x 4 = 8
- 3 x (-3) = -9
- (-1) x 1 = -1
- 8 + (-9) + (-1) = -2
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The result of the dot product is -2 and this result is written at the top left of the matrix product.
To find the term in the top right of the matrix product, just look for the dot product of the top row of Matrix A and the right column of Matrix B. Here's how you do it:
- 2 x (-5) = -10
- 3 x 0 = 0
- (-1) x 2 = -2
- -10 + 0 + (-2) = -12
- The result of the dot product is -12 and this result is written at the top right of the matrix product.
Step 6. Make sure that the four dot products are in the correct place in the matrix product
19 must be at the bottom left, -34 must be at the bottom right, -2 must be at the top left, and -12 must be at the top right.
Tips
- Using line segments, and not using lines, can give the wrong answer. If a line representing a row requires extension to cross a column, then lengthen it! This is just a visualization technique to make it easier for you to know which rows and columns to use to work with each element of the product.
- The product of the two matrices will produce the number of rows equal to the number of rows of the first matrix and the number of columns equal to the number of columns of the second matrix.
- Write down your sum. Multiplying matrices involves a lot of calculations and it's very easy to get sidetracked and forget which number you're multiplying.
