How to Multiply Matrix? (+FREE Worksheet!)
How to multiply matrix, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
Multiply Matrix: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Check dimensionsRows by columns determines what operation is legal.
- Use the correct ruleAddition is entry-by-entry; multiplication is row-by-column.
- Interpret the resultFor systems, translate the matrix answer back into variables.
Worked examples
Check if multiplication is allowed
- Compare the inner dimensions.
- The 3 and 3 match.
- Use the outer dimensions for the answer size.
Compute one entry
- Multiply matching row-column entries.
- 2 times 3 is 6 and 5 times 4 is 20.
- Add the products.
Try one before moving on
Multiply Matrix: pop-up practice
Matrix Multiplication, Example 1:
(begin{bmatrix}-5 & -5 -1 & 2 end{bmatrix})(begin{bmatrix}-2 & -3 3 & 5 end{bmatrix})
Solution:
Multiply the rows of the first matrix by the columns of the second matrix. (begin{bmatrix}(-5)(-2)+(-5)(3) & (-5)(-3)+(-5)(5) (-1)(-2)+(2)(3) & (-1)(-3)+(2)(5) end{bmatrix}= begin{bmatrix}(10)+(-15) & (15)+(-25) (2)+(6) & (3)+(10) end{bmatrix}=begin{bmatrix}-5 & -10 8 & 13 end{bmatrix})
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Matrix Multiplication, Example 2:
\(\begin{bmatrix}-4 & -6 & -6 \\ 0 & 6 & 3\end{bmatrix}\begin{bmatrix}0 \\ -3 \\ 0\end{bmatrix}\)
Solution:
Multiply each row by the column: the first entry is (-4)(0) + (-6)(-3) + (-6)(0) = 18. The second is (0)(0) + (6)(-3) + (3)(0) = -18. The product is \(\begin{bmatrix}18 \\ -18\end{bmatrix}\).
Matrix Multiplication, Example 3:
(begin{bmatrix}1 & 3 2 & 4 end{bmatrix})(begin{bmatrix}2 &4 -2 & 1 end{bmatrix})
Solution:
(begin{bmatrix}(1) (2)+(3)(-2) & (1) (4)+(3) (1) (2) (2)+ (4)(-2) & (2) (4)+(4) (1) end{bmatrix}=begin{bmatrix}(2)+(-6) & (4)+(3) (4)+ (-8) & (8)+(4) end{bmatrix}=begin{bmatrix}-4 & 7 -4 & 12 end{bmatrix})
Matrix Multiplication, Example 4:
(begin{bmatrix}2 & -1&-1 3 & 1&5 end{bmatrix}begin{bmatrix}-2 -1 4 end{bmatrix})
Solution:
Multiply the rows of the first matrix by the columns of the second matrix. (begin{bmatrix}(2)(-2)+(-1)(-1)+(-1) (4)(3)(-2)+(1)(-1)+(5) (4) end{bmatrix}=begin{bmatrix}(-4)+(1)+(-4)(-6)+(-1)+(20) end{bmatrix}=begin{bmatrix}-7 13 end{bmatrix})
Exercises for Multiplying Matrix
Solve.
- (color{blue}{begin{bmatrix}0 & 2 -2 & -5 end{bmatrix}begin{bmatrix}6 & -6 3 & 0 end{bmatrix}})
- (color{blue}{begin{bmatrix}3 & -1 -3 & 6-6&-6 end{bmatrix}begin{bmatrix}-1 & 6 5 & 4end{bmatrix}})
- (color{blue}{begin{bmatrix}0 & 5 -3 & 1-5&1 end{bmatrix}begin{bmatrix}-4 & 4 -2 & -4end{bmatrix}})
- (color{blue}{begin{bmatrix}5 & 3&5 1 & 5&0 end{bmatrix}begin{bmatrix}-4 & 2 -3 & 43&-5 end{bmatrix}})
- (color{blue}{begin{bmatrix}4 & 5 -4 & 6-5&-6 end{bmatrix}begin{bmatrix}4 & 6 6& 2-4&1 end{bmatrix}})
- (color{blue}{begin{bmatrix}-2 & -6 -4 & 35&0 4&-6end{bmatrix}begin{bmatrix}2 & -2&2 -2 &0&-3 end{bmatrix}})
- (color{blue}{begin{bmatrix}6 & 0 -27 & 12 end{bmatrix}})
- (color{blue}{begin{bmatrix}-8 & 14 33 & 6 -24&-60end{bmatrix}})
- (color{blue}{begin{bmatrix}-10 & -20 10 & -16 18&-24end{bmatrix}})
- (color{blue}{begin{bmatrix}-14 & -3 -19 & 22 end{bmatrix}})
- (color{blue}{Undefined})
- (color{blue}{begin{bmatrix}8 & 4&14-14 & 8&-1710&-10&10 20&-8&26end{bmatrix}})
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