$\left[\begin{array}{cc}-2 & -6 \\ -4 & 3 \\ 5 & 0 \\ 4 & -6\end{array}\right] \cdot\left[\begin{array}{ccc}2 & -2 & 2 \\ -2 & 0 & -3\end{array}\right]$
Answer (2)
Multiply the first row of the first matrix by the first, second, and third columns of the second matrix to find the first row of the resulting matrix.
Multiply the second row of the first matrix by the first, second, and third columns of the second matrix to find the second row of the resulting matrix.
Multiply the third row of the first matrix by the first, second, and third columns of the second matrix to find the third row of the resulting matrix.
Multiply the fourth row of the first matrix by the first, second, and third columns of the second matrix to find the fourth row of the resulting matrix.
The resulting matrix is [ 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 elements of the resulting matrix.
Calculating the Elements 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.
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 The resulting 4x3 matrix is:
[ 8 4 14 − 14 8 − 17 10 − 10 10 20 − 8 26 ]
Examples
Matrix multiplication is a fundamental operation in computer graphics, where it's used to perform transformations on 3D models. For instance, if you have a 3D model represented as a set of vertices and you want to rotate, scale, or translate it, you can represent these transformations as matrices. By multiplying the model's vertex matrix by the transformation matrix, you can efficiently update the model's position and orientation in space. This is crucial for rendering realistic scenes in video games, simulations, and other graphical applications.
To multiply the given matrices, we find the dot products of the rows from the first matrix with the columns from the second matrix. The resulting matrix is a 4x3 matrix with computed elements. The final resulting matrix is 8 − 14 10 20 4 8 − 10 − 8 14 − 17 10 26 .
;
