if [tex]A=\left[\begin{array}{ll}2 & 3 \\ 10 & 15\end{array}\right][/tex] and [tex]B\left[\begin{array}{ll}4 & 7 \\ 8 & 24\end{array}\right][/tex] find [tex]A+B[/tex], [tex]A-B[/tex], [tex]2A+3B[/tex].
Answer (2)
The calculations show that 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 .
Subtract corresponding elements of B from A to find A − B .
Multiply A by 2 and B by 3, then add the resulting matrices to find 2 A + 3 B .
The results are: A + B = [ 6 18 10 39 ] , A − B = [ − 2 2 − 4 − 9 ] , and 2 A + 3 B = [ 16 44 27 102 ] .
Explanation
Understanding the Problem 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 involve adding or subtracting corresponding elements. 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 used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, matrices are used to represent transformations such as scaling, rotation, and translation of objects in 3D space. By performing matrix addition and multiplication, we can combine multiple transformations into a single matrix, which can then be applied to the object's vertices to render the transformed object on the screen. This allows for efficient and complex manipulations of graphical objects.
