Question

How do you find \[2A-3B\] given \[A=\left( 5\ \ -2\ \ 3\ \ 1 \right)\] and \[B=\left( -2\ \ 3\ \ 1\ \ 0 \right)\] ?

Answer 1

Hint: From the question given we have to find \[2A-3B\] for the given matrices \[A=\left( 5\ \ -2\ \ 3\ \ 1 \right)\] and \[B=\left( -2\ \ 3\ \ 1\ \ 0 \right)\] . The given two matrices have the same columns and rows so without any doubt we can add and subtract them easily. \[2A\] Means multiplying each element present in the matrix A with the number \[2\] and \[3B\] means multiplying each and every element with the number \[3\] . After doing the multiplication we can subtract them directly then we will get a matrix that is the required answer.

Complete step by step solution:
From the question we have two matrices they are,
\[\Rightarrow A=\left( 5\ \ -2\ \ 3\ \ 1 \right)\]
\[\Rightarrow B=\left( -2\ \ 3\ \ 1\ \ 0 \right)\]
By observing the two matrices the columns and rows of the two matrices are equal i.e., the columns of A and B are \[4\], and the rows of A and B are \[\ 1\] . a column in a matrix means vertical lines and row of the matrix means horizontal lines.
By this we can conclude that we can perform addition and subtraction properties on the given matrices.
Now, we have to find \[2A-3B\].
\[2A\] means we have to multiply matrix A with \[2\].
By multiplying matrix, A with \[2\] we get,
\[\Rightarrow A=\left( 5\ \ -2\ \ 3\ \ 1 \right)\]
\[\Rightarrow 2A=\left( 2\times 5\ \ 2\times -2\ \ 2\times 3\ \ 2\times 1 \right)\]
\[\Rightarrow 2A=\left( 10\ \ -4\ \ 6\ \ 2 \right)\]
 \[3B\] means multiplying each and every element with the number \[3\].
By multiplying matrix, B with \[3\] we get,
\[\Rightarrow B=\left( -2\ \ 3\ \ 1\ \ 0 \right)\]
\[\Rightarrow 3B=\left( 3\times -2\ \ 3\times 3\ \ 3\times 1\ \ 3\times 0 \right)\]
\[\Rightarrow 3B=\left( -6\ \ 9\ \ 3\ \ 0 \right)\]
Now we have to find \[2A-3B\] by doing this we will get,
\[\Rightarrow 2A-3B\]
\[\Rightarrow 2A-3B=\left( 10\ \ -4\ \ 6\ \ 2 \right)-\left( -6\ \ 9\ \ 3\ \ 0 \right)\]
\[\Rightarrow 2A-3B=\left( 10-\left( -6 \right)\ \ -4-9\ \ 6-3\ \ 2-0 \right)\]
\[\Rightarrow 2A-3B=\left( 16\ \ -13\ \ 3\ \ 2 \right)\]
Therefore, the required answer is \[2A-3B=\left( 16\ \ -13\ \ 3\ \ 2 \right)\].

Note: Students should recall properties of matrices while doing the above problem. Students should also know that while doing multiplication of two matrices the columns of the first matrix should be equal to rows of the second matrix if this condition satisfies students should proceed with the multiplication of two matrices.