If [tex]$A=\left[\begin{array}{lll}3 & 1 & 5 \\ 2 & 6 & 4 \\ 8 & 7 & 9\end{array}\right]$[/tex] and [tex]$B=\left[\begin{array}{lll}1 & 2 & 4 \\ 1 & 5 & 3 \\ 2 & 4 & 0\end{array}\right]$[/tex], find [tex]$5A$[/tex] and [tex]$-2A+3B$[/tex].
Answer (2)
We computed 5 A and − 2 A + 3 B using the matrices A and B. The results are 5 A = 15 10 40 5 30 35 25 20 45 and − 2 A + 3 B = − 3 − 1 − 10 4 3 − 2 2 1 − 18 .
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Multiply each element of matrix A by 5 to find 5A: 5 A = 15 10 40 5 30 35 25 20 45 .
Multiply each element of matrix A by -2 to find -2A: − 2 A = − 6 − 4 − 16 − 2 − 12 − 14 − 10 − 8 − 18 .
Multiply each element of matrix B by 3 to find 3B: 3 B = 3 3 6 6 15 12 12 9 0 .
Add the corresponding elements of -2A and 3B to find -2A + 3B: − 2 A + 3 B = − 3 − 1 − 10 4 3 − 2 2 1 − 18 .
Explanation
Problem Analysis We are given two matrices, A and B , and we need to compute 5 A and − 2 A + 3 B . This involves scalar multiplication and matrix addition.
Calculating 5A First, let's find 5 A . We multiply each element of matrix A by 5:
5 A = 5 × 3 2 8 1 6 7 5 4 9 = 5 × 3 5 × 2 5 × 8 5 × 1 5 × 6 5 × 7 5 × 5 5 × 4 5 × 9 = 15 10 40 5 30 35 25 20 45
Calculating -2A Next, let's find − 2 A . We multiply each element of matrix A by -2:
− 2 A = − 2 × 3 2 8 1 6 7 5 4 9 = − 2 × 3 − 2 × 2 − 2 × 8 − 2 × 1 − 2 × 6 − 2 × 7 − 2 × 5 − 2 × 4 − 2 × 9 = − 6 − 4 − 16 − 2 − 12 − 14 − 10 − 8 − 18
Calculating 3B Now, let's find 3 B . We multiply each element of matrix B by 3:
3 B = 3 × 1 1 2 2 5 4 4 3 0 = 3 × 1 3 × 1 3 × 2 3 × 2 3 × 5 3 × 4 3 × 4 3 × 3 3 × 0 = 3 3 6 6 15 12 12 9 0
Calculating -2A + 3B Finally, let's find − 2 A + 3 B . We add the corresponding elements of the matrices − 2 A and 3 B :
− 2 A + 3 B = − 6 − 4 − 16 − 2 − 12 − 14 − 10 − 8 − 18 + 3 3 6 6 15 12 12 9 0 = − 6 + 3 − 4 + 3 − 16 + 6 − 2 + 6 − 12 + 15 − 14 + 12 − 10 + 12 − 8 + 9 − 18 + 0 = − 3 − 1 − 10 4 3 − 2 2 1 − 18
Final Answer Therefore, 5 A = 15 10 40 5 30 35 25 20 45 and − 2 A + 3 B = − 3 − 1 − 10 4 3 − 2 2 1 − 18 .
Examples
Matrix operations, like the ones we just performed, are fundamental in computer graphics. When you see a 3D object rotating or scaling on your screen, the transformations are often done using matrices. Scalar multiplication changes the size of the object, while matrix addition combines different transformations to create complex animations. These concepts are also used in image processing, where matrices can represent images and operations like blurring or sharpening can be achieved through matrix manipulations.
