How to multiply a 3x3 matrix by a 3x1 matrix?
Understand the Problem
The question is asking for the process of multiplying a 3x3 matrix by a 3x1 matrix. This involves taking the dot product of the rows of the 3x3 matrix with the single column of the 3x1 matrix to produce a new 3x1 matrix as a result.
Answer
The resulting matrix $C$ is given by: $$ C = \begin{bmatrix} a_{11}b_{1} + a_{12}b_{2} + a_{13}b_{3} \\ a_{21}b_{1} + a_{22}b_{2} + a_{23}b_{3} \\ a_{31}b_{1} + a_{32}b_{2} + a_{33}b_{3} \end{bmatrix} $$
Answer for screen readers
The resulting 3x1 matrix $C$ is given by: $$ C = \begin{bmatrix} a_{11}b_{1} + a_{12}b_{2} + a_{13}b_{3} \ a_{21}b_{1} + a_{22}b_{2} + a_{23}b_{3} \ a_{31}b_{1} + a_{32}b_{2} + a_{33}b_{3} \end{bmatrix} $$
Steps to Solve
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Identify the matrices Determine your 3x3 matrix and 3x1 matrix. Let the 3x3 matrix be denoted as: $$ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{bmatrix} $$ And the 3x1 matrix as: $$ B = \begin{bmatrix} b_{1} \ b_{2} \ b_{3} \end{bmatrix} $$
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Multiply the first row Calculate the first element of the resulting 3x1 matrix by taking the dot product of the first row of matrix $A$ with matrix $B$: $$ R_1 = a_{11} \cdot b_{1} + a_{12} \cdot b_{2} + a_{13} \cdot b_{3} $$
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Multiply the second row Calculate the second element of the resulting 3x1 matrix by taking the dot product of the second row of matrix $A$ with matrix $B$: $$ R_2 = a_{21} \cdot b_{1} + a_{22} \cdot b_{2} + a_{23} \cdot b_{3} $$
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Multiply the third row Calculate the third element of the resulting 3x1 matrix by taking the dot product of the third row of matrix $A$ with matrix $B$: $$ R_3 = a_{31} \cdot b_{1} + a_{32} \cdot b_{2} + a_{33} \cdot b_{3} $$
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Combine the results The final resulting 3x1 matrix $C$ is formed by combining the three results: $$ C = \begin{bmatrix} R_1 \ R_2 \ R_3 \end{bmatrix} $$
The resulting 3x1 matrix $C$ is given by: $$ C = \begin{bmatrix} a_{11}b_{1} + a_{12}b_{2} + a_{13}b_{3} \ a_{21}b_{1} + a_{22}b_{2} + a_{23}b_{3} \ a_{31}b_{1} + a_{32}b_{2} + a_{33}b_{3} \end{bmatrix} $$
More Information
When you multiply a matrix by another matrix, each element of the resulting matrix is computed as the sum of products, as shown above. Matrix multiplication is widely used in computer graphics, data science, and equations of physics.
Tips
- Mixing up the order of matrix multiplication. Remember, matrix multiplication is not commutative; $AB \neq BA$ in general.
- Forgetting to multiply all the corresponding elements correctly.
