\(\left[\begin{array}{cc}3 & -1 \\ -3 & 6 \\ -6 & -6\end{array}\right] \cdot\left[\begin{array}{cc}-1 & 6 \\ 5 & 4\end{array}\right]\)
Answer (1)
Multiply the first row of the first matrix by the first column of the second matrix to get the element (1,1): ( 3 × − 1 ) + ( − 1 × 5 ) = − 8 .
Multiply the first row of the first matrix by the second column of the second matrix to get the element (1,2): ( 3 × 6 ) + ( − 1 × 4 ) = 14 .
Multiply the second row of the first matrix by the first column of the second matrix to get the element (2,1): ( − 3 × − 1 ) + ( 6 × 5 ) = 33 .
Multiply the second row of the first matrix by the second column of the second matrix to get the element (2,2): ( − 3 × 6 ) + ( 6 × 4 ) = 6 .
Multiply the third row of the first matrix by the first column of the second matrix to get the element (3,1): ( − 6 × − 1 ) + ( − 6 × 5 ) = − 24 .
Multiply the third row of the first matrix by the second column of the second matrix to get the element (3,2): ( − 6 × 6 ) + ( − 6 × 4 ) = − 60 .
The resulting matrix is: [ − 8 14 33 6 − 24 − 60 ] .
Explanation
Problem Analysis We are asked to multiply two matrices. The first matrix is a 3x2 matrix, and the second matrix is a 2x2 matrix. The product of these matrices will be a 3x2 matrix.
Method Description 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.
Calculations Let's calculate the elements of the resulting 3x2 matrix:
Element (1,1): ( 3 × − 1 ) + ( − 1 × 5 ) = − 3 − 5 = − 8 Element (1,2): ( 3 × 6 ) + ( − 1 × 4 ) = 18 − 4 = 14 Element (2,1): ( − 3 × − 1 ) + ( 6 × 5 ) = 3 + 30 = 33 Element (2,2): ( − 3 × 6 ) + ( 6 × 4 ) = − 18 + 24 = 6 Element (3,1): ( − 6 × − 1 ) + ( − 6 × 5 ) = 6 − 30 = − 24 Element (3,2): ( − 6 × 6 ) + ( − 6 × 4 ) = − 36 − 24 = − 60
Final Result Therefore, the resulting matrix is: [ − 8 14 33 6 − 24 − 60 ]
Examples
Matrix multiplication is used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, matrices are used to represent transformations such as rotation, scaling, and translation of objects in 3D space. Multiplying a matrix representing an object's vertices by a transformation matrix applies the transformation to the object. This allows for efficient manipulation and rendering of complex scenes.
