[\begin{array}{ll}
1 & 2 \\
2 & 5 \\
4 & 0
\end{array}] \cdot[\begin{array}{ll}
3 & 7 \\
0 & 2
\end{array}]=[\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}]
Answer (1)
Calculate a 11 = ( 1 ) ( 3 ) + ( 2 ) ( 0 ) = 3 .
Calculate a 12 = ( 1 ) ( 7 ) + ( 2 ) ( 2 ) = 11 .
Calculate a 21 = ( 2 ) ( 3 ) + ( 5 ) ( 0 ) = 6 .
Calculate a 22 = ( 2 ) ( 7 ) + ( 5 ) ( 2 ) = 24 .
Calculate a 31 = ( 4 ) ( 3 ) + ( 0 ) ( 0 ) = 12 .
Calculate a 32 = ( 4 ) ( 7 ) + ( 0 ) ( 2 ) = 28 .
The resulting matrix is [ 3 11 6 24 12 28 ] .
Explanation
Understanding the Problem We are given a matrix multiplication problem. Our goal is to find the elements of the resulting matrix.
Calculating a_11 To find the element a 11 , we multiply the first row of the first matrix by the first column of the second matrix: a 11 = ( 1 ) ( 3 ) + ( 2 ) ( 0 ) = 3 + 0 = 3 .
Calculating a_12 To find the element a 12 , we multiply the first row of the first matrix by the second column of the second matrix: a 12 = ( 1 ) ( 7 ) + ( 2 ) ( 2 ) = 7 + 4 = 11 .
Calculating a_21 To find the element a 21 , we multiply the second row of the first matrix by the first column of the second matrix: a 21 = ( 2 ) ( 3 ) + ( 5 ) ( 0 ) = 6 + 0 = 6 .
Calculating a_22 To find the element a 22 , we multiply the second row of the first matrix by the second column of the second matrix: a 22 = ( 2 ) ( 7 ) + ( 5 ) ( 2 ) = 14 + 10 = 24 .
Calculating a_31 To find the element a 31 , we multiply the third row of the first matrix by the first column of the second matrix: a 31 = ( 4 ) ( 3 ) + ( 0 ) ( 0 ) = 12 + 0 = 12 .
Calculating a_32 To find the element a 32 , we multiply the third row of the first matrix by the second column of the second matrix: a 32 = ( 4 ) ( 7 ) + ( 0 ) ( 2 ) = 28 + 0 = 28 .
Final Result Therefore, the resulting matrix is [ 3 11 6 24 12 28 ] .
Examples
Matrix multiplication is used in computer graphics to perform transformations such as scaling, rotation, and translation of objects. For example, if you have a 3D model represented as a set of vertices, you can use matrix multiplication to rotate the model by a certain angle. Each vertex of the model is represented as a vector, and the rotation is represented as a matrix. By multiplying the rotation matrix by each vertex vector, you can obtain the new coordinates of the vertices after the rotation. This is a fundamental concept in creating realistic 3D graphics.
