Given matrices [tex]A[/tex] and [tex]B[/tex], find the number in row 1, column 2 of [tex]B \cdot A[/tex].

[tex]
A=\left[\begin{array}{llll}
4 & 2 & 3 & 6 \\
4 & 1 & 0 & 3
\end{array}\right] \quad B=\left[\begin{array}{llll}
5 & 4 & 6 & 0 \\
1 & 1 & 2 & 0 \\
3 & 3 & 0 & 1
\end{array}\right]
[/tex]

A. Can't be done
B. 10
C. 8
D. 24



Answer :

To solve this problem, we need to determine if the matrix multiplication [tex]\( B \cdot A \)[/tex] is possible and then find the entry in row 1, column 2 of the resulting matrix.

1. Determine the dimensions of the matrices:

Matrix [tex]\( A \)[/tex] is a [tex]\( 2 \times 4 \)[/tex] matrix:
[tex]\[ A = \begin{pmatrix} 4 & 2 & 3 & 6 \\ 4 & 1 & 0 & 3 \\ \end{pmatrix} \][/tex]

Matrix [tex]\( B \)[/tex] is a [tex]\( 3 \times 4 \)[/tex] matrix:
[tex]\[ B = \begin{pmatrix} 5 & 4 & 6 & 0 \\ 1 & 1 & 2 & 0 \\ 3 & 3 & 0 & 1 \\ \end{pmatrix} \][/tex]

2. Check if the matrix multiplication [tex]\( B \cdot A \)[/tex] is possible:

For matrix multiplication [tex]\( B \cdot A \)[/tex] to be defined, the number of columns in [tex]\( B \)[/tex] must equal the number of rows in [tex]\( A \)[/tex].

- Matrix [tex]\( B \)[/tex] has 4 columns.
- Matrix [tex]\( A \)[/tex] has 2 rows.

Since the number of columns in [tex]\( B \)[/tex] (which is 4) does not match the number of rows in [tex]\( A \)[/tex] (which is 2), the matrix multiplication [tex]\( B \cdot A \)[/tex] is not possible.

Therefore, the problem cannot be solved as [tex]\( B \cdot A \)[/tex] cannot be done.

The final answer is:
[tex]\[ \text{can't be done} \][/tex]