Find \(\frac{1}{2} C\).
\(C=\left[\begin{array}{ll}
12 & -20 \\
36 & -52
\end{array}\right]\)
Answer (1)
The problem asks to find 2 1 C where C is a given matrix.
Multiply each element of the matrix C by the scalar 2 1 .
Calculate the new elements: 2 1 × 12 = 6 , 2 1 × − 20 = − 10 , 2 1 × 36 = 18 , 2 1 × − 52 = − 26 .
The resulting matrix is [ 6 − 10 18 − 26 ] .
Explanation
Understanding the Problem We are given a matrix C and asked to find 2 1 C . This means we need to multiply each element of the matrix C by the scalar 2 1 .
Stating the Matrix The given matrix is: C = [ 12 − 20 36 − 52 ]
Scalar Multiplication To find 2 1 C , we multiply each element of C by 2 1 :
2 1 C = [ 2 1 × 12 2 1 × − 20 2 1 × 36 2 1 × − 52 ]
Calculating the Elements Now, we perform the calculations: 2 1 × 12 = 6 2 1 × − 20 = − 10 2 1 × 36 = 18 2 1 × − 52 = − 26 So, the resulting matrix is: 2 1 C = [ 6 − 10 18 − 26 ]
Final Answer Therefore, the final answer is: 2 1 C = [ 6 − 10 18 − 26 ]
Examples
Scalar multiplication of matrices is used in various fields such as computer graphics, physics, and engineering. For example, in computer graphics, scaling an object involves multiplying the coordinates of its vertices by a scalar. If you have a square defined by the matrix [[2, 0], [0, 2]], multiplying this matrix by 0.5 would shrink the square to half its size, resulting in the matrix [[1, 0], [0, 1]]. This technique is fundamental in resizing and manipulating images and 3D models.
