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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 $2A$.

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

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

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