Question

Find the value for the matrix $ \left[ {\begin{array}{*{20}{c}}
{ - 1}&4&2
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
5\\
1\\
3
\end{array}} \right] $ .

Answer 1

Hint: When we are multiplying two matrices, then first, multiply the first row with the first column and substitute in the first. Now, multiply the first row with the second column and substitute the in the second. Multiply the second row with the first column and substitute in three and multiply the second row with the second column to substitute the obtained value in the fourth.

Complete step-by-step answer:
Given:
The matrix is $ \left[ {\begin{array}{*{20}{c}}
{ - 1}&4&2
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
5\\
1\\
3
\end{array}} \right] $ .

We know that for the multiplication of matrices the number of columns of the first matrix must be equal to the number of rows of the second matrix. Then after multiplication the resultant matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. The multiplication is a binary operation.

Here, In our problem, the first matrix is $ 1 \times 3 $ and the second matrix is $ 3 \times 1 $ .
Then after multiplication we get the matrix is $ \left( {1 \times 3} \right) \times \left( {3 \times 1} \right) \to 1 \times 1 $ .
If the two matrices are of the form $ m \times n $ and $ p \times k $ .
We can multiply matrices only when $ n = p $ .
In the multiplication of the matrix we will first multiply $ - 1 $ with $ 5 $ and add with the product of $ 4 $ and $ 1 $ . Then add the product of the $ 2 $ and $ 3 $ .

Therefore, the product for the given matrix $ \left[ {\begin{array}{*{20}{c}}
{ - 1}&4&2
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
5\\
1\\
3
\end{array}} \right] $ is $ \left[ {\begin{array}{*{20}{c}}
{ - 1}&4&2
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
5\\
1\\
3
\end{array}} \right] = \left[ { - 5 + 4 + 6} \right] = \left[ 5 \right] $ .

Therefore, the value for the product of two matrices is $ 5 $ .

Note: If one of the matrices have rows as $ m $ and column as $ n $ and the other matrix is of the matrix which is having rows as $ k $ and column as $ p $ . If $ k \ne n $ then we cannot multiply two matrices. If $ k = n $ then we can multiply the two matrices.