Multiply the following matrices:
[tex]A=\left|\begin{array}{llll}
2 & 4 & 0 & 1 \\
7 & 0 & 2 & 2
\end{array}\right| \quad B=\left|\begin{array}{lll}
2 & 5 & 1 \\
0 & 7 & 1 \\
2 & 0 & 0 \\
7 & 2 & 0
\end{array}\right|[/tex]
Answer (2)
Multiply the first row of matrix A by the first column of matrix B to get the element (1,1) of the resulting matrix: ( 2 × 2 ) + ( 4 × 0 ) + ( 0 × 2 ) + ( 1 × 7 ) = 11 .
Multiply the first row of matrix A by the second column of matrix B to get the element (1,2) of the resulting matrix: ( 2 × 5 ) + ( 4 × 7 ) + ( 0 × 0 ) + ( 1 × 2 ) = 40 .
Multiply the second row of matrix A by the first column of matrix B to get the element (2,1) of the resulting matrix: ( 7 × 2 ) + ( 0 × 0 ) + ( 2 × 2 ) + ( 2 × 7 ) = 32 .
The resulting matrix is [ 11 32 40 39 6 7 ] .
Explanation
Understanding the Problem We are asked to multiply two matrices, A and B. Matrix A is a 2x4 matrix, and matrix B is a 4x3 matrix. Therefore, the resulting matrix will be a 2x3 matrix.
Matrix Multiplication Rule 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 matrix A and the j-th column of matrix B.
Calculating the Elements Let's calculate each element of the resulting 2x3 matrix:
Element (1,1): ( 2 × 2 ) + ( 4 × 0 ) + ( 0 × 2 ) + ( 1 × 7 ) = 4 + 0 + 0 + 7 = 11
Element (1,2): ( 2 × 5 ) + ( 4 × 7 ) + ( 0 × 0 ) + ( 1 × 2 ) = 10 + 28 + 0 + 2 = 40
Element (1,3): ( 2 × 1 ) + ( 4 × 1 ) + ( 0 × 0 ) + ( 1 × 0 ) = 2 + 4 + 0 + 0 = 6
Element (2,1): ( 7 × 2 ) + ( 0 × 0 ) + ( 2 × 2 ) + ( 2 × 7 ) = 14 + 0 + 4 + 14 = 32
Element (2,2): ( 7 × 5 ) + ( 0 × 7 ) + ( 2 × 0 ) + ( 2 × 2 ) = 35 + 0 + 0 + 4 = 39
Element (2,3): ( 7 × 1 ) + ( 0 × 1 ) + ( 2 × 0 ) + ( 2 × 0 ) = 7 + 0 + 0 + 0 = 7
Final Result Therefore, the resulting matrix is:
[ 11 32 40 39 6 7 ]
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. In economics, matrix multiplication can be used to analyze the relationships between different sectors of an economy.
The resulting matrix from multiplying matrices A and B is [ 11 32 40 39 6 7 ] . This was calculated by taking the dot product of rows from A with columns from B. Each element was derived step-by-step by multiplying respective entries and summing them up.
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