Question

Let A and B be m × n and r × s matrices respectively. Use summation notation to define (ab)ij , the ijth entry of the product AB. Explain carefully why we need n = r for this definition to make sense.

Answer 1

Matrix are group of numbers represented using rows and columns.

The reason why we need $\mathbf{n = r}$ is that: The rows of matrix A would be multiplied by the columns of matrix B;

From the question, we have:

$\mathbf{A = m \times n}$

$\mathbf{B = r \times s}$

When the product of both matrices is taken, the dimension of the resulting matrix is: $\mathbf{AB = m \times s}$

The above dimension of AB means that: $\mathbf{n = r}$

The possible interpretations are:

• The column of A must-have the same value has the row of B
• The columns of A would be multiplied to the rows of B, to calculate AB

So, the meaning of $\mathbf{(AB)_{ij}}$ is that

The elements in the i-th row of matrix A, are multiplied by the elements in the j-th column of B

Hence, the reason why we need $\mathbf{n = r}$ is that:

The rows of matrix A would be multiplied by the columns of matrix B; so they need to be equal.

Read more about matrix at:

https://brainly.com/question/9967572