13.1: Basic Matrix Multiplication
- Page ID
- 85044
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By Carey Smith
Matrix Multiplication Summary:
A matrix times a vector (3x3 case):
\( A*B =
\begin{matrix}
|a & b & c| \\
|p & q & r| \\
|u & v & w|
\end{matrix} * \begin{matrix}
|x| \\
|y| \\
|z|
\end{matrix} = \begin{matrix}
|a*x & b*y & c*z| \\
|p*x & q*y & r*z| \\
|u*x & v*y & w*z|
\end{matrix}
\)
Matrices do not need to be square, but the number of columns in A needs to equal the number of rows in B.
Matrix multiplication is generally not commutative. That is A*B ≠ B*A, in general.
.
Watch this video about multiplying matrices:
Matlab Video Tutorial: Multiplying Matrices and Vectors, https://www.youtube.com/watch?v=sgzFn42jU6Y
.
A = [3 2 1;
2 1 0
1 0 -1]
B = [4
5
6]
A*B
Solution
A*B =
28
13
-2
% = [28 %(28 = 3*4 + 2*5 + 1*6)
% 13 %(13 = 2*4 + 1*5 + 0*6)
% -2] %(-2 = 1*4 + 0*5 + -1*6)
.
1. True or False: These 2 matrices (which have 3 rows and 2 columns) can be multiplied together:
[ 1 5 ] [ 9 5 ]
A = | 2 9 | B = | 8 9 |
[ 3 4 ] [ 7 5 ]
2. True or False: These 2 matrices can be multiplied together:
[ 1 5 ]
A = | 2 9 |; B = [ 1 3 2 ]
[ 3 4 ] [ 4 6 9 ]
- Answer
-
1. No, because the number of columns in A (2) does not equal the number of rows in B (3).
1. Yes, because the number of columns in A (2) does equal the number of rows in B (2).
Matrix Multiplication Calculation
1. What is A*B ?
A = [1 4
3 -5]
B = [ 2 12
3 1]
(Do a matrix multiply, not an element-by element multiply)
- Answer
-
A*B = [14 16-9 31]
.
A matrix with ones on the diagonal and zeros elsewhere is called the Identity matrix:
I = eye(3) = [1 0 0
0 1 0
0 0 1]
I*A = A
A*I = A
For more information, see: https://en.Wikipedia.org/wiki/Matrix_multiplication
.