$\left[\begin{array}{cc}0 & 5 \\ -3 & 1 \\ -5 & 1\end{array}\right] \cdot\left[\begin{array}{cc}-4 & 4 \\ -2 & -4\end{array}\right]$
Answer (1)
Multiply the first row of the first matrix by the first column of the second matrix to get the element (1,1) of the resulting matrix: ( 0 ) ( − 4 ) + ( 5 ) ( − 2 ) = − 10 .
Multiply the first row of the first matrix by the second column of the second matrix to get the element (1,2) of the resulting matrix: ( 0 ) ( 4 ) + ( 5 ) ( − 4 ) = − 20 .
Multiply the second row of the first matrix by the first column of the second matrix to get the element (2,1) of the resulting matrix: ( − 3 ) ( − 4 ) + ( 1 ) ( − 2 ) = 10 .
Multiply the second row of the first matrix by the second column of the second matrix to get the element (2,2) of the resulting matrix: ( − 3 ) ( 4 ) + ( 1 ) ( − 4 ) = − 16 .
Multiply the third row of the first matrix by the first column of the second matrix to get the element (3,1) of the resulting matrix: ( − 5 ) ( − 4 ) + ( 1 ) ( − 2 ) = 18 .
Multiply the third row of the first matrix by the second column of the second matrix to get the element (3,2) of the resulting matrix: ( − 5 ) ( 4 ) + ( 1 ) ( − 4 ) = − 24 .
The resulting matrix is [ − 10 − 20 10 − 16 18 − 24 ] .
Explanation
Understanding the Problem We are asked to multiply two matrices. The first matrix is a 3x2 matrix, and the second matrix is a 2x2 matrix.
Checking Matrix Dimensions To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, the first matrix has 2 columns and the second matrix has 2 rows, so we can multiply them. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. So the resulting matrix will be a 3x2 matrix.
Calculating the Elements The element in the i-th row and j-th column of the resulting matrix is obtained by taking the dot product of the i-th row of the first matrix and the j-th column of the second matrix. Let's calculate the elements of the resulting 3x2 matrix:
c 11 = ( 0 ) ( − 4 ) + ( 5 ) ( − 2 ) = 0 − 10 = − 10 c 12 = ( 0 ) ( 4 ) + ( 5 ) ( − 4 ) = 0 − 20 = − 20 c 21 = ( − 3 ) ( − 4 ) + ( 1 ) ( − 2 ) = 12 − 2 = 10 c 22 = ( − 3 ) ( 4 ) + ( 1 ) ( − 4 ) = − 12 − 4 = − 16 c 31 = ( − 5 ) ( − 4 ) + ( 1 ) ( − 2 ) = 20 − 2 = 18 c 32 = ( − 5 ) ( 4 ) + ( 1 ) ( − 4 ) = − 20 − 4 = − 24
Writing the Resulting Matrix So the resulting matrix is:
[ − 10 − 20 10 − 16 18 − 24 ]
Final Answer The product of the two matrices is [ − 10 − 20 10 − 16 18 − 24 ] .
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. In physics, matrices are used to represent linear transformations, such as rotations and reflections, and to solve systems of linear equations. In engineering, matrices are used in structural analysis to determine the stresses and strains in a structure under load.
