Question

Let C=[[-6,0],[1,2]] and D=[[-6,4,7],[5,-7,6]] Find CD if it is defined.

Answer 1

To multiply matrices C and D, we compute the dot product for each position in the resulting matrix. Since the dimensions are compatible, the matrix multiplication is defined, resulting in a 2x3 matrix CD with computed values.

Step-by-step explanation:

To find the product of two matrices, in this case CD, we must follow the matrix multiplication rules. Each element of the resulting matrix is computed as the dot product of the corresponding row from the first matrix C and the column from the second matrix D.

Matrix C is a 2x2 matrix, and matrix D is a 2x3 matrix. According to the rules of matrix multiplication, the number of columns in the first matrix (C) must equal the number of rows in the second matrix (D). In this case, the number of columns in C (which is 2) is equal to the number of rows in D (which is also 2), so matrix multiplication is defined.

The product CD will result in a new 2x3 matrix. Here's how it's calculated:

1. The element in position (1,1) of matrix CD: (-6)*(-6) + (0)*(5) = 36 + 0 = 36
2. The element in position (1,2) of matrix CD: (-6)*(4) + (0)*(-7) = -24 + 0 = -24
3. The element in position (1,3) of matrix CD: (-6)*(7) + (0)*(6) = -42 + 0 = -42
4. The element in position (2,1) of matrix CD: (1)*(-6) + (2)*(5) = -6 + 10 = 4
5. The element in position (2,2) of matrix CD: (1)*(4) + (2)*(-7) = 4 - 14 = -10
6. The element in position (2,3) of matrix CD: (1)*(7) + (2)*(6) = 7 + 12 = 19

Therefore, the product matrix CD is:

CD = [[36, -24, -42],[4, -10, 19]]