Find $2A-3B$ if $A = \begin{bmatrix} 17 & 5 & 19 \\ 11 & 8 & 13 \end{bmatrix} $ and $B = \begin{bmatrix} 9 & 3 & 7 \\ 1 & 6 & 5 \end{bmatrix} $

$A =
\begin{bmatrix}
17 & 5 & 19 \\
11 & 8 & 13
\end{bmatrix}
$ and $B =
\begin{bmatrix}
9 & 3 & 7 \\
1 & 6 & 5
\end{bmatrix}
$ are two matrices. It is required to find the value of an expression $2A-3B$. Actually, the two matrices are $2 \times 3$. It is possible and can be evaluated by some basic mathematical operations in matrices.

Multiply matrix A by 2

The value of $2A$ can be obtained by multiplying the matrix $A$ by number $2$. It can be done by multiplying every element in the matrix $A$ by $2$.

$2 \times A \,=\,
2 \times \begin{bmatrix}
17 & 5 & 19 \\
11 & 8 & 13
\end{bmatrix}$

$\implies 2A \,=\,
2 \times \begin{bmatrix}
17 & 5 & 19 \\
11 & 8 & 13
\end{bmatrix}$

$\implies 2A \,=\,
\begin{bmatrix}
2 \times 17 & 2 \times 5 & 2 \times 19 \\
2 \times 11 & 2 \times 8 & 2 \times 13
\end{bmatrix}$

$\,\,\, \therefore \,\,\,\,\,\, 2A \,=\,
\begin{bmatrix}
34 & 10 & 38 \\
22 & 16 & 26
\end{bmatrix}$

Multiply matrix B by 3

Similarly, the value of $3B$ can be evaluated by multiplying the matrix $B$ by number $3$. It can also be done by multiplying each element in the matrix $B$ by $3$.

$3 \times B \,=\,
3 \times \begin{bmatrix}
9 & 3 & 7 \\
1 & 6 & 5
\end{bmatrix}$

$\implies 3B \,=\,
3 \times \begin{bmatrix}
9 & 3 & 7 \\
1 & 6 & 5
\end{bmatrix}$

$\implies 3B \,=\,
\begin{bmatrix}
3 \times 9 & 3 \times 3 & 3 \times 7 \\
3 \times 1 & 3 \times 6 & 3 \times 5
\end{bmatrix}$

$\,\,\, \therefore \,\,\,\,\,\, 3B \,=\,
\begin{bmatrix}
27 & 9 & 21 \\
3 & 18 & 15
\end{bmatrix}$

Subtract 3B matrix from 2A

The values of $2A$ and $3B$ are two matrices. The value of $2A-3B$ can be obtained by subtracting the matrix $3B$ from the matrix $2A$.

$2A-3B$ $\,=\,$ $\begin{bmatrix}
34 & 10 & 38 \\
22 & 16 & 26
\end{bmatrix}$ $\,-\,$ $\begin{bmatrix}
27 & 9 & 21 \\
3 & 18 & 15
\end{bmatrix}$

$\implies 2A-3B$ $\,=\,$ $\begin{bmatrix}
34-27 & 10-9 & 38-21 \\
22-3 & 16-18 & 26-15
\end{bmatrix}$

$\,\,\, \therefore \,\,\,\,\,\, 2A-3B$ $\,=\,$ $\begin{bmatrix}
7 & 1 & 17 \\
19 & -2 & 11
\end{bmatrix}$