How to Multiply Matrices

Reviewed by Joseph Meyer

Last Updated: September 20, 2022 References

A matrix is a rectangular arrangement of numbers, symbols, or expressions in rows and columns. To multiply matrices, you'll need to multiply the elements (or numbers) in the row of the first matrix by the elements in the rows of the second matrix and add their products. You can multiply matrices in just a few easy steps that require addition, multiplication, and the proper placement of the results.

Steps

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1
Confirm that the matrices can be multiplied. You can only multiply matrices if the number of columns of the first matrix is equal to the number of rows in the second matrix.^{[1]
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- These matrices can be multiplied because the first matrix, Matrix A, has 3 columns, while the second matrix, Matrix B, has 3 rows.
2
Mark the dimensions of the matrix product. Create a new blank matrix that will mark the dimensions of the matrix product, 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 indicate the number of rows and columns in this matrix.^{[2]
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- Matrix A has 2 rows, so the matrix product will have 2 rows.
- Matrix B has 2 columns, so the matrix product will have 2 columns.
- The matrix product will have 2 rows and 2 columns.
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3
Find the first dot product. To find a dot product, you need to multiply the first element in the first row by the first element of the first column, the second element of the first row by the second element of the first column, and the third element in the first row by the third element in the first column. Then, add their products to find the dot product.^{[3]
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Research source} Let's say you've decided to solve for the element in the 2^nd row and 2^nd column (bottom right) of the matrix product first. Here's how you do it:^{[4]
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- 6 x -5 = -30
- 1 x 0 = 0
- -2 x 2 = -4
- -30 + 0 + (-4) = -34
- The dot product is -34 and it belongs on the bottom right of the matrix product.
  - When you multiply matrices, the dot product will go in the position of the row of the first Matrix and the column of the second matrix.^{[5]
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    Research source} For example, when you found the dot product of the bottom row of Matrix A and the right column of Matrix B, the answer, -34, went in the bottom row and right column of the matrix product.
4
Find the second dot product. Let's say you want to find the term on the bottom left of the matrix product. To find this term, you simply have to multiply the elements on the bottom row of the first matrix with the elements in the first column of the second matrix and then add them up. Use the same method you used to multiply the first row and column -- find the dot product again.^{[6]
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- 6 x 4 = 24
- 1 x (-3) = -3
- (-2) x 1 = -2
- 24 + (-3) + (-2) = 19
- The dot product is -19 and it belongs on the bottom left of the matrix product.
5
Find the remaining two dot products. To find the term on the top 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.^{[7]
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Research source} Here's how you do it:
- 2 x 4 = 8
- 3 x (-3) = -9
- (-1) x 1 = -1
- 8 + (-9) + (-1) = -2
- The dot product is -2 and it belongs on the top left of the matrix product.
  - To find the term on the top right of the matrix product, just find 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 dot product is -12 and it belongs on the top right of the matrix product.
19 should be in the bottom left, -34 should be on the bottom right, -2 should be on the top left, and -12 should be on the top right.^{[8]
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How can I find the inverse with my final answer?

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This article should be helpful: how to find the inverse of a function.

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How do I multiply a 2x1 matrix with a 2x2 matrix?

Pimemorized

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You cannot multiply a 2x1 matrix with a 2x2 matrix together. To multiply two matrices together, the first matrix's columns and the second matrix's rows have to be the same. In this case, the first matrix only has 1 column, whereas the second one has two rows.

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How do I multiply a 1×3 matrix by a 3×1 matrix?

I_l1ke_gam3s

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Use the dot product shown in the article above. (Note: You cannot switch the order and get the same answer, as matrix multiplication isn't commutative.)

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Tips

Write out your sums. Multiplying matrices involves a lot of calculations and it is very easy to get distracted and lose track of which numbers you are multiplying.

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Using line segments instead of lines may result in wrong answers. If the line representing a row needs extending to cross a column then extend it! This is just a visualisation technique to make it easier to work out which row and column should be used to work out each element of the product.

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The product of the two matrices should have the same number of rows as first matrix and the same number of columns as the second matrix.

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