If [tex]A=\left[\begin{array}{cc}2 & 3 \\ 10 & 15\end{array}\right][/tex] and [tex]B=\left[\begin{array}{cc}4 & 7 \\ 8 & 24\end{array}\right][/tex], find [tex]A+B[/tex], [tex]A-B[/tex], [tex]2A+3B[/tex], and [tex]B[/tex].
Answer (2)
The results of the matrix operations are: A + B = [ 6 18 10 39 ] , A − B = [ − 2 2 − 4 − 9 ] , and 2 A + 3 B = [ 16 44 27 102 ] .
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Add corresponding elements of A and B to find A + B = [ 6 18 10 39 ] .
Subtract corresponding elements of B from A to find A − B = [ − 2 2 − 4 − 9 ] .
Multiply A by 2 and B by 3, then add the results to find 2 A + 3 B = [ 16 44 27 102 ] .
The final results are: A + B = [ 6 18 10 39 ] , A − B = [ − 2 2 − 4 − 9 ] , and 2 A + 3 B = [ 16 44 27 102 ] .
Explanation
Problem Analysis We are given two matrices, A and B , and we need to compute A + B , A − B , and 2 A + 3 B . Matrix addition and subtraction are performed element-wise. Scalar multiplication involves multiplying each element of the matrix by the scalar.
Calculating A+B To find A + B , we add the corresponding elements of A and B :
A + B = [ 2 10 3 15 ] + [ 4 8 7 24 ] = [ 2 + 4 10 + 8 3 + 7 15 + 24 ] = [ 6 18 10 39 ]
Calculating A-B To find A − B , we subtract the corresponding elements of B from A :
A − B = [ 2 10 3 15 ] − [ 4 8 7 24 ] = [ 2 − 4 10 − 8 3 − 7 15 − 24 ] = [ − 2 2 − 4 − 9 ]
Calculating 2A+3B To find 2 A + 3 B , we first multiply A by 2 and B by 3: 2 A = 2 [ 2 10 3 15 ] = [ 4 20 6 30 ] 3 B = 3 [ 4 8 7 24 ] = [ 12 24 21 72 ] Then, we add the resulting matrices: 2 A + 3 B = [ 4 20 6 30 ] + [ 12 24 21 72 ] = [ 4 + 12 20 + 24 6 + 21 30 + 72 ] = [ 16 44 27 102 ]
Final Answer Therefore, we have: A + B = [ 6 18 10 39 ] A − B = [ − 2 2 − 4 − 9 ] 2 A + 3 B = [ 16 44 27 102 ]
Examples
Matrix operations are fundamental in various fields such as computer graphics, physics, and engineering. For instance, in computer graphics, matrices are used to represent transformations such as scaling, rotation, and translation of objects in 3D space. By performing matrix addition, subtraction, and scalar multiplication, we can manipulate these transformations to achieve desired visual effects. In physics, matrices are used to describe systems of linear equations, which arise in many areas such as circuit analysis and quantum mechanics. Understanding matrix operations is crucial for solving these equations and analyzing the behavior of physical systems.
