Question

question 1 - 1
a matrix equation is shown
\begin{bmatrix} 4& - 5&0\\3&1&2\\ - 7&6&4 end{bmatrix}\times\begin{bmatrix} 9& - 3&6\\ - 2&8&7\\3& - 1&5 end{bmatrix}=\begin{bmatrix} a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} end{bmatrix}
enter the value of (a_{31}) in the box
(a_{31}=)

Explanation:

Step1: Recall matrix - multiplication rule

To find the element \(a_{21}\) of the product matrix, we use the formula for matrix - multiplication. The element in the \(i\) - th row and \(j\) - th column of the product matrix \(C = A\times B\) is given by \(c_{ij}=\sum_{k = 1}^{n}a_{ik}b_{kj}\), where \(n\) is the number of columns of matrix \(A\) (or rows of matrix \(B\)). Here, to find \(a_{21}\), we consider the second row of the first matrix and the first column of the second matrix.

Step2: Calculate \(a_{21}\)

The second row of the first matrix is \([3\ 1\ 2]\), and the first column of the second matrix is \(

$$\begin{bmatrix}9\\ - 2\\3\end{bmatrix}$$

\). Then \(a_{21}=3\times9 + 1\times(-2)+2\times3\).
\[

$$\begin{align*} a_{21}&=27-2 + 6\\ &=31 \end{align*}$$

\]

Answer:

\(31\)