Example Question - elements

Matrix Multiplication Result

The image shows two matrices that need to be multiplied. Let's call the first matrix A and the second matrix B: Matrix A = [3 2 5] [2 3 1] Matrix B = [4 5 -5] [5 -1 6] [4 5 -5] To multiply two matrices, we can use the following formula for the elements of the resulting matrix: Given matrices A (of size m by n) and B (of size n by p), their product C (of size m by p) will have elements: \[ c_{ij} = \sum_{k=1}^{n} a_{ik} \cdot b_{kj} \] So for matrices A (2x3) and B (3x3), we can find the product C (2x3) by multiplying each row of A with each column of B. Let's perform the multiplication: \[ c_{11} = (3 \times 4) + (2 \times 5) + (5 \times 4) = 12 + 10 + 20 = 42 \] \[ c_{12} = (3 \times 5) + (2 \times -1) + (5 \times 5) = 15 - 2 + 25 = 38 \] \[ c_{13} = (3 \times -5) + (2 \times 6) + (5 \times -5) = -15 + 12 - 25 = -28 \] \[ c_{21} = (2 \times 4) + (3 \times 5) + (1 \times 4) = 8 + 15 + 4 = 27 \] \[ c_{22} = (2 \times 5) + (3 \times -1) + (1 \times 5) = 10 - 3 + 5 = 12 \] \[ c_{23} = (2 \times -5) + (3 \times 6) + (1 \times -5) = -10 + 18 - 5 = 3 \] So the product C of the matrices A and B is: Matrix C = [42 38 -28] [27 12 3]