Question

How do you multiply a $3\times 3$ matrix by a $3\times 1$ matrix?

Answer 1

Hint:To perform multiplication of two matrices, we should make sure that the number of columns in the $1\;st$ matrix is equal to the rows in the $2\;nd$ matrix.Therefore, the resulting matrix product will have a number of rows of the $1\;st$ matrix and a number of columns of the $2\;nd$ matrix. The order of the resulting matrix is the matrix multiplication order.

Complete step by step answer:
Matrix multiplication, also known as matrix product and the multiplication of two matrices, produces a single matrix. It is a type of binary operation.If $A$ and $B$ are the two matrices, then the product of the two matrices A and B are denoted by: $X = AB$. Hence, the product of two matrices is the dot product of the two matrices.

Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative,even when the product remains definite after changing the order of the factors.

Before we multiply two matrices, we have two ensure that the number of columns in the first matrix is equal to the number of rows in another matrix.Here we have to multiply the $3\times 3$ matrix and $3\times 1$ matrix, which is possible and the resultant matrix will be $3\times 1$.For example :
Let $A = \left( {\begin{array}{*{20}{c}}
1&4&6 \\
2&3&0 \\
0&7&0
\end{array}} \right)$ and $B = \left( {\begin{array}{*{20}{c}}
8 \\
4 \\
0
\end{array}} \right)$
$AB = \left( {\begin{array}{*{20}{c}}
{(1).(2) + (4).(0) + (6).(1)} \\
{(2).(2) + (3).(0) + (0).(1)} \\
{(0).(2) + (7).(0) + (0).(1)}
\end{array}} \right)$
$\Rightarrow AB= \left( {\begin{array}{*{20}{c}}
8 \\
4 \\
0
\end{array}} \right)$

Note:The product of two matrices $A$ and $B$ is defined if the number of columns of $A$ is equal to the number of rows of $B$ . If $AB\;$ is defined, then $BA\;$ need not be defined. If both $A$ and $B$ are square matrices of the same order, then both $AB\;$ and $BA\;$ are defined. If $AB\;$ and $BA\;$ are both defined, it is not necessary that $AB = BA$. If the product of two matrices is a zero matrix, it is not necessary that one of the matrices is a zero matrix.