Calculate the following matrix products.
\begin{pmatrix} 11 & 12 & 13 \cr\cr 21 & 22 & 23 \end{pmatrix}\times\begin{pmatrix} 11 \cr\cr 21 \cr\cr 31 \end{pmatrix}
Let A=\begin{matrix}\left(a_{ij}\right)\end{matrix} be a n\times p matrix and B=\begin{matrix}\left(b_{ij}\right)\end{matrix} be a p\times m matrix. The product C=AB is a n\times m matrix with entries c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}.
Note the number of columns in A must equal the number of rows in B to have a matrix product.
Here the product of a 2\times3 matrix with a 3\times1 matrix is a 2\times1 matrix.
We have:
\begin{pmatrix} 11 & 12 & 13 \cr\cr 21 & 22 & 23 \end{pmatrix}\times\begin{pmatrix} 11 \cr\cr 21 \cr\cr 31 \end{pmatrix}=\begin{pmatrix} 11 \times 11+12\times21+13\times31 \cr\cr 21\times 11+22 \times 21+23\times 31 \end{pmatrix}
\begin{pmatrix} 11 & 12 & 13 \cr\cr 21 & 22 & 23 \end{pmatrix}\times\begin{pmatrix} 11 \cr\cr 21 \cr\cr 31 \end{pmatrix}=\begin{pmatrix} 121+252+403\cr\cr 231+462+713 \end{pmatrix}
\begin{pmatrix} 11 & 12 & 13 \cr\cr 21 & 22 & 23 \end{pmatrix}\times\begin{pmatrix} 11 \cr\cr 21 \cr\cr 31 \end{pmatrix}=\begin{pmatrix} 776 \cr\cr 1\ 406 \end{pmatrix}
\begin {pmatrix} 1 & 3& 2 \cr \cr 1 & 1 & 5 \end {pmatrix} \times \begin {pmatrix} 2 \cr \cr 2 \cr \cr 3 \end {pmatrix}
Let A=\begin{matrix}\left(a_{ij}\right)\end{matrix} be a n\times p matrix and B=\begin{matrix}\left(b_{ij}\right)\end{matrix} be a p\times m matrix. The product C=AB is a n\times m matrix with entries c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}.
Note the number of columns in A must equal the number of rows in B to have a matrix product.
Here the product of a 2\times3 matrix with a 3\times1 matrix is a 2\times1 matrix.
We have:
\begin {pmatrix} 1 & 3& 2 \cr \cr 1 & 1 & 5 \end {pmatrix} \times \begin {pmatrix} 2 \cr \cr 2 \cr \cr 3 \end {pmatrix}=\begin{pmatrix} 1 \times 2+3\times2+2\times3 \cr\cr 1\times 2+1 \times 2+5\times 3 \end{pmatrix}
\begin {pmatrix} 1 & 3& 2 \cr \cr 1 & 1 & 5 \end {pmatrix} \times \begin {pmatrix} 2 \cr \cr 2 \cr \cr 3 \end {pmatrix}=\begin{pmatrix} 14\cr\cr 19 \end{pmatrix}
\begin {pmatrix} 1 & 3& 2 \cr \cr 1 & 1 & 5 \end {pmatrix} \times \begin {pmatrix} 2 \cr \cr 2 \cr \cr 3 \end {pmatrix}=\begin{pmatrix} 14\cr\cr 19 \end{pmatrix}
\begin {pmatrix} 5& 4& 1 \cr \cr 7 & 2 & 1 \end {pmatrix} \times \begin {pmatrix} 1 \cr \cr 2 \cr \cr 1 \end {pmatrix}
Let A=\begin{matrix}\left(a_{ij}\right)\end{matrix} be a n\times p matrix and B=\begin{matrix}\left(b_{ij}\right)\end{matrix} be a p\times m matrix. The product C=AB is a n\times m matrix with entries c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}.
Note the number of columns in A must equal the number of rows in B to have a matrix product.
Here the product of a 2\times3 matrix with a 3\times1 matrix is a 2\times1 matrix.
We have:
\begin {pmatrix} 5& 4& 1 \cr \cr 7 & 2 & 1 \end {pmatrix} \times \begin {pmatrix} 1 \cr \cr 2 \cr \cr 1 \end {pmatrix}=\begin{pmatrix} 5\times 1+4\times2+1 \times 1 \cr\cr 7\times1+2\times2+1\times1 \end{pmatrix}
\begin {pmatrix} 5& 4& 1 \cr \cr 7 & 2 & 1 \end {pmatrix} \times \begin {pmatrix} 1 \cr \cr 2 \cr \cr 1 \end {pmatrix}=\begin{pmatrix} 14\cr 12\end{pmatrix}
\begin {pmatrix} 5& 4& 1 \cr \cr 7 & 2 & 1 \end {pmatrix} \times \begin {pmatrix} 1 \cr \cr 2 \cr \cr 1 \end {pmatrix}=\begin{pmatrix} 14\cr 12\end{pmatrix}
\begin{pmatrix} 2 & 5 \cr 1 & 6\end{pmatrix}\times \begin{pmatrix} 6 \cr 3 \end{pmatrix}
Let A=\begin{matrix}\left(a_{ij}\right)\end{matrix} be a n\times p matrix and B=\begin{matrix}\left(b_{ij}\right)\end{matrix} be a p\times m matrix. The product C=AB is a n\times m matrix with entries c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}.
Note the number of columns in A must equal the number of rows in B to have a matrix product.
Here the product of a 2\times2 matrix with a 2\times1 matrix is a 2\times1 matrix.
We have:
\begin{pmatrix} 2 & 5 \cr 1 & 6\end{pmatrix}\times \begin{pmatrix} 6 \cr 3 \end{pmatrix}=\begin{pmatrix} 2\times 6+5\times 3 \cr 1 \times 6 +6 \times 3\end{pmatrix}
\begin{pmatrix} 2 & 5 \cr 1 & 6\end{pmatrix}\times \begin{pmatrix} 6 \cr 3 \end{pmatrix}=\begin{pmatrix} 27 \cr 24\end{pmatrix}
\begin{pmatrix} 2 & 5 \cr 1 & 6\end{pmatrix}\times \begin{pmatrix} 6 \cr 3 \end{pmatrix}=\begin{pmatrix} 27 \cr 24\end{pmatrix}
\begin{pmatrix} 1 & 4 \cr 2 & 7\end{pmatrix}\times \begin{pmatrix} 3 & 2 \cr 3 & 1 \end{pmatrix}
Let A=\begin{matrix}\left(a_{ij}\right)\end{matrix} be a n\times p matrix and B=\begin{matrix}\left(b_{ij}\right)\end{matrix} be a p\times m matrix. The product C=AB is a n\times m matrix with entries c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}.
Note the number of columns in A must equal the number of rows in B to have a matrix product.
Here the product of a 2\times2 matrix with a 2\times2 matrix is a 2\times2 matrix.
We have:
\begin{pmatrix} 1 & 4 \cr 2 & 7\end{pmatrix}\times \begin{pmatrix} 3 & 2 \cr 3 & 1 \end{pmatrix}=\begin{pmatrix} 1\times 3+4\times3 & 1\times2+4\times1 \cr 2\times3+7\times 3 & 2\times2+7\times 1\end{pmatrix}
\begin{pmatrix} 1 & 4 \cr 2 & 7\end{pmatrix}\times \begin{pmatrix} 3 & 2 \cr 3 & 1 \end{pmatrix}=\begin{pmatrix} 15 & 6 \cr 27 & 11\end{pmatrix}
\begin{pmatrix} 1 & 4 \cr 2 & 7\end{pmatrix}\times \begin{pmatrix} 3 & 2 \cr 3 & 1 \end{pmatrix}=\begin{pmatrix} 15 & 6 \cr 27 & 11\end{pmatrix}
\begin{pmatrix} 2 & 3 \cr 4 & 1\end{pmatrix}\times \begin{pmatrix} 1 & 4 \cr -1 & -2 \end{pmatrix}
Let A=\begin{matrix}\left(a_{ij}\right)\end{matrix} be a n\times p matrix and B=\begin{matrix}\left(b_{ij}\right)\end{matrix} be a p\times m matrix. The product C=AB is a n\times m matrix with entries c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}.
Note the number of columns in A must equal the number of rows in B to have a matrix product.
Here the product of a 2\times2 matrix with a 2\times2 matrix is a 2\times2 matrix.
We have:
\begin{pmatrix} 2 & 3 \cr 4 & 1\end{pmatrix}\times \begin{pmatrix} 1 & 4 \cr -1 & -2 \end{pmatrix}=\begin{pmatrix} 2\times 1+3\times -1 & 2\times 4+3\times -2 \cr 4\times 1+1\times -1 & 4\times 4+1\times -2\end{pmatrix}
\begin{pmatrix} 2 & 3 \cr 4 & 1\end{pmatrix}\times \begin{pmatrix} 1 & 4 \cr -1 & -2 \end{pmatrix}=\begin{pmatrix} -1 & 2 \cr 3 & 14\end{pmatrix}
\begin{pmatrix} 2 & 3 \cr 4 & 1\end{pmatrix}\times \begin{pmatrix} 1 & 4 \cr -1 & -2 \end{pmatrix}=\begin{pmatrix} -1 & 2 \cr 3 & 14\end{pmatrix}
\begin{pmatrix} 1 & 2 \cr 7 & 11\end{pmatrix}\times \begin{pmatrix} 11 & -4 \cr -1 & 2 \end{pmatrix}
Let A=\begin{matrix}\left(a_{ij}\right)\end{matrix} be a n\times p matrix and B=\begin{matrix}\left(b_{ij}\right)\end{matrix} be a p\times m matrix. The product C=AB is a n\times m matrix with entries c_{ij}=\sum_{k=1}^{p}a_{ik}b_{kj}.
Note the number of columns in A must equal the number of rows in B to have a matrix product.
Here the product of a 2\times2 matrix with a 2\times2 matrix is a 2\times2 matrix.
We have:
\begin{pmatrix} 1 & 2 \cr 7 & 11\end{pmatrix}\times \begin{pmatrix} 11 & -4 \cr -1 & 2 \end{pmatrix}=\begin{pmatrix} 1\times 11+2\times -1 & 1\times -4+2\times 2 \cr 7\times 11+11\times -1 & 7\times -4+11\times 2\end{pmatrix}
\begin{pmatrix} 1 & 2 \cr 7 & 11\end{pmatrix}\times \begin{pmatrix} 11 & -4 \cr -1 & 2 \end{pmatrix}=\begin{pmatrix} 9 & 0 \cr 66 & -6\end{pmatrix}
\begin{pmatrix} 1 & 2 \cr 7 & 11\end{pmatrix}\times \begin{pmatrix} 11 & -4 \cr -1 & 2 \end{pmatrix}=\begin{pmatrix} 9 & 0 \cr 66 & -6\end{pmatrix}