In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix}-4 & -4\\-3 &-1 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} 4\\ -1\end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} -16+4\\-12+1 \end{vmatrix}=\begin{vmatrix} -12\\-11 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix}-6 & 0\\1 & -2\end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} -1 \\ 0 \end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} 6+0\\-1+0 \end{vmatrix}=\begin{vmatrix} 6\\-1 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix} 5& 7\\9 &-5 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} -2\\-8 \end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix}-10-56 \\ -18+40\end{vmatrix}=\begin{vmatrix} -66\\22 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix} 4& -9\\6 &9 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} 6\\5 \end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} 24-45\\36+45 \end{vmatrix}=\begin{vmatrix} -21\\81 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix} -9&-3 \\2 &6 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} 3\\ -8\end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} -27+24\\6-48 \end{vmatrix}=\begin{vmatrix} -3\\-42 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that

\(\displaystyle A=\begin{vmatrix} -8& -9\\6 &3 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} 7\\-4 \end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} -56+36\\42-12 \end{vmatrix}=\begin{vmatrix} -20\\30 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix} 4&-9 \\-2 &-9 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} -6\\-3 \end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} -24+27\\ 12+27\end{vmatrix}=\begin{vmatrix} 3\\39 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix} 6&-9 \\6 &-1 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} 2\\ 3\end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} 12-27\\ 12-3\end{vmatrix}=\begin{vmatrix} -15\\9 \end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that

\(\displaystyle A=\begin{vmatrix} -9&-1 \\4 &-5 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix}-9\\-5 \end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix}81+5 \\-36+25 \end{vmatrix}=\begin{vmatrix} 86\\ -11\end{vmatrix}\)

In order to multiply two matrices, \(\displaystyle A\times b\), the respective dimensions of each must be of the form \(\displaystyle m \times n\) and \(\displaystyle n \times p\) to create an \(\displaystyle m\times p\) (notation is rows x columns) matrix. Unlike the multiplication of individual values, the order of the matrices does matter.

\(\displaystyle (A\times b \neq b \times A)\)

For a multiplication of the form

\(\displaystyle \begin{vmatrix} A_{1,1}&A_{1,2} &... &A_{1,n} \\ A_{2,1}&A_{2,2} &.. &A_{2,n} \\ ...& ...&... &... \\ A_{m,1}&A_{m,2} &... &A_{m,n} \end{vmatrix}\times \begin{vmatrix} b_{1,1}&b_{1,2} &... &b_{1,p} \\ b_{2,1}&b_{2,2} &... &b_{2,p} \\ ...&... &... &... \\ b_{n,1}&b_{n,2} &... &b_{n,p} \end{vmatrix}\)

The resulting matrix is

\(\displaystyle \begin{vmatrix} A_{1,1}b_{1,1}+A_{1,2}b_{2,1}+...+A_{1,n}b_{n,1}&... &A_{1,1}b_{1,p}+A_{1,2}b_{2,p}+...+A_{1,n}b_{n,p} \\ ...&... &... \\ A_{m,1}b_{1,1}+A_{m,2}b_{2,1}+...+A_{m,n}b_{n,1}&... &A_{m,1}b_{1,p}+A_{m,2}b_{2,p}+...+A_{m,n}b_{n,p} \end{vmatrix}\)

The notation may be daunting but numerical examples may elucidate.

We're told that 

\(\displaystyle A=\begin{vmatrix} 0&-8 \\-7 &8 \end{vmatrix}\) and \(\displaystyle b=\begin{vmatrix} 5\\5 \end{vmatrix}\)

The resulting matrix product is then:

\(\displaystyle A\times b = \begin{vmatrix} 0-40\\-35+40 \end{vmatrix}=\begin{vmatrix} -40\\5 \end{vmatrix}\)