$A=\left[\begin{array}{ll} 2 & 3 \\ 10 & 15 \end{array}\right] \text { and } B\left[\begin{array}{ll} 4 & 7 \\ 8 & 24 \end{array}\right]$. Find $A+B$, $A-B$, $2A+3B$.
Answer (2)
We calculated the matrices resulting from addition, subtraction, and scalar multiplication. The results are: A + B = \begin{bmatrix} 6 & 10 \ 18 & 39 \end{bmatrix}, A - B = \begin{bmatrix} -2 & -4 \ 2 & -9 \end{bmatrix}, and 2A + 3B = \begin{bmatrix} 16 & 27 \ 44 & 102 \end{bmatrix}.
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Calculate A + B by adding corresponding elements: A + B = [ 6 18 10 39 ] .
Calculate A − B by subtracting corresponding elements: A − B = [ − 2 2 − 4 − 9 ] .
Calculate 2 A and 3 B : 2 A = [ 4 20 6 30 ] and 3 B = [ 12 24 21 72 ] .
Calculate 2 A + 3 B by adding the resulting matrices: 2 A + 3 B = [ 16 44 27 102 ] .
Explanation
Understanding the Problem We are given two matrices, A and B , and we need to find A + B , A − B , and 2 A + 3 B . Matrix addition and subtraction involve adding or subtracting corresponding elements in the matrices. 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 A and B :
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 need to find 2 A and 3 B . To find 2 A , we multiply each element of A by 2: 2 A = 2 × [ 2 10 3 15 ] = [ 2 × 2 2 × 10 2 × 3 2 × 15 ] = [ 4 20 6 30 ] To find 3 B , we multiply each element of B by 3: 3 B = 3 × [ 4 8 7 24 ] = [ 3 × 4 3 × 8 3 × 7 3 × 24 ] = [ 12 24 21 72 ] Now, we add 2 A and 3 B :
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. Adding matrices can combine transformations, while scalar multiplication can scale objects. These operations are fundamental to rendering and manipulating images and models in games and simulations.
