Multiply the matrices, then enter the value of $A_{21}$ of the resulting matrix.
$\left[\begin{array}{cc}
-3 & 4 \\
2 & 7
\end{array}\right]\left[\begin{array}{ll}
0 & 2 \\
1 & 0
\end{array}\right]=A$
Answer (1)
Multiply the given matrices: [ − 3 4 2 7 ] [ 0 2 1 0 ] = [ 4 − 6 7 4 ] .
Identify the element A 21 in the resulting matrix.
A 21 is the element in the second row and first column, which is 7.
The value of A 21 is 7 .
Explanation
Problem Setup and Matrix Definition We are given two matrices and asked to multiply them and then identify the element in the second row and first column of the resulting matrix. Let's call the first matrix M 1 and the second matrix M 2 . So, we have
M 1 = [ − 3 4 2 7 ] , M 2 = [ 0 2 1 0 ]
We want to find the matrix A = M 1 M 2 and then identify the element A 21 .
Matrix Multiplication To find the matrix A , we perform matrix multiplication. The element in the i -th row and j -th column of A is obtained by taking the dot product of the i -th row of M 1 and the j -th column of M 2 .
A = [ ( − 3 ) ( 0 ) + ( 4 ) ( 1 ) ( − 3 ) ( 2 ) + ( 4 ) ( 0 ) ( 2 ) ( 0 ) + ( 7 ) ( 1 ) ( 2 ) ( 2 ) + ( 7 ) ( 0 ) ]
A = [ 0 + 4 − 6 + 0 0 + 7 4 + 0 ]
A = [ 4 − 6 7 4 ]
Identifying the Element A21 Now we need to identify the element A 21 of the matrix A . This is the element in the second row and first column, which is 7.
Final Answer Therefore, the value of A 21 is 7.
7
Examples
Matrix multiplication is a fundamental operation in computer graphics. For example, when rendering a 3D scene, matrices are used to represent transformations such as rotation, scaling, and translation of objects. By multiplying a series of transformation matrices together, we can efficiently apply multiple transformations to an object with a single matrix multiplication. The element A 21 could represent a specific component of a transformation, such as the scaling factor in the x-direction after a series of rotations and shears. Understanding matrix multiplication is crucial for creating realistic and interactive visual experiences.
