$\left[\begin{array}{cc}-2 & -6 \\ -4 & 3 \\ 5 & 0 \\ 4 & -6\end{array}\right] \cdot\left[\begin{array}{ccc}2 & -2 & 2 \\ -2 & 0 & -3\end{array}\right]$
Answer (1)
[ 8 4 14 − 14 8 − 17 10 − 10 10 20 − 8 26 ]
Explanation
Understanding the Problem We are asked to multiply two matrices. The first matrix is a 4x2 matrix, and the second matrix is a 2x3 matrix. Therefore, the resulting matrix will be a 4x3 matrix. We will perform matrix multiplication to find the product.
Setting up Matrix Multiplication To find the element in the i-th row and j-th column of the resulting matrix, we take the dot product of the i-th row of the first matrix and the j-th column of the second matrix. Let's calculate each element:
Calculating the Elements Element (1,1): ( − 2 ) ( 2 ) + ( − 6 ) ( − 2 ) = − 4 + 12 = 8
Element (1,2): ( − 2 ) ( − 2 ) + ( − 6 ) ( 0 ) = 4 + 0 = 4
Element (1,3): ( − 2 ) ( 2 ) + ( − 6 ) ( − 3 ) = − 4 + 18 = 14
Element (2,1): ( − 4 ) ( 2 ) + ( 3 ) ( − 2 ) = − 8 − 6 = − 14
Element (2,2): ( − 4 ) ( − 2 ) + ( 3 ) ( 0 ) = 8 + 0 = 8
Element (2,3): ( − 4 ) ( 2 ) + ( 3 ) ( − 3 ) = − 8 − 9 = − 17
Element (3,1): ( 5 ) ( 2 ) + ( 0 ) ( − 2 ) = 10 + 0 = 10
Element (3,2): ( 5 ) ( − 2 ) + ( 0 ) ( 0 ) = − 10 + 0 = − 10
Element (3,3): ( 5 ) ( 2 ) + ( 0 ) ( − 3 ) = 10 + 0 = 10
Element (4,1): ( 4 ) ( 2 ) + ( − 6 ) ( − 2 ) = 8 + 12 = 20
Element (4,2): ( 4 ) ( − 2 ) + ( − 6 ) ( 0 ) = − 8 + 0 = − 8
Element (4,3): ( 4 ) ( 2 ) + ( − 6 ) ( − 3 ) = 8 + 18 = 26
Final Result Therefore, the resulting matrix is: [ 8 4 14 − 14 8 − 17 10 − 10 10 20 − 8 26 ]
Examples
Matrix multiplication is used in various fields such as computer graphics, physics, and economics. For example, in computer graphics, matrices are used to represent transformations such as rotation, scaling, and translation of objects in 3D space. By multiplying matrices, we can combine multiple transformations into a single matrix, which can then be applied to the object's vertices to produce the final image. This is a fundamental concept in creating realistic and interactive 3D environments.
