L-A First Course in Linear Algebra

Chapter M: Matrices

Section MM

Page 190

C21

Compute the product AB of the two matrices below using both the definition of the matrix-vector product (Definition MVP) and the definition of matrix multiplication (Definition MM).

\[\mathbf{A} = \left[\begin{array}{rrr}1 & 3 & 2 \\-1 & 2 & 1 \\0 & 1 & 0\end{array}\right] \mathbf{B} = \left[\begin{array}{rrr}4 & 1 & 2 \\1 & 0 &1 \\3 & 1 & 5\end{array}\right]\]

Solution using R
##      [,1] [,2] [,3]
## [1,]    1    3    2
## [2,]   -1    2    1
## [3,]    0    1    0
##      [,1] [,2] [,3]
## [1,]    4    1    2
## [2,]    1    0    1
## [3,]    3    1    5
##      [,1] [,2] [,3]
## [1,]   13    3   15
## [2,]    1    0    5
## [3,]    1    0    1
(a) Using MVP (matrix-vector product)

\[ \mathbf{A*B} = \left[\begin{array}{rrr}B\end{array}\right]_1A_1 + \left[\begin{array}{rrr}B\end{array}\right]_2A_2 + \left[\begin{array}{rrr}B\end{array}\right]_3A_3 \] \[ = \left[\begin{array}{rrr}4\left[\begin{array}{rrr}1 \\-1 \\0\end{array}\right] + 1\left[\begin{array}{rrr}3 \\2 \\1\end{array}\right] + 3\left[\begin{array}{rrr}2 \\1 \\0\end{array}\right] |1\left[\begin{array}{rrr}1 \\-1 \\0\end{array}\right] + 0\left[\begin{array}{rrr}3 \\2 \\1\end{array}\right] + 1\left[\begin{array}{rrr}2 \\1 \\0\end{array}\right] | 2\left[\begin{array}{rrr}1 \\-1 \\0\end{array}\right] + 1\left[\begin{array}{rrr}3 \\2 \\1\end{array}\right] + 5\left[\begin{array}{rrr}2 \\1 \\0\end{array}\right] \end{array}\right] \] \[ = \left[\begin{array}{rrr}\left[\begin{array}{rrr}4 \\-4 \\0\end{array}\right] + \left[\begin{array}{rrr}3 \\2 \\1\end{array}\right] + \left[\begin{array}{rrr}6 \\3 \\0\end{array}\right] |\left[\begin{array}{rrr}1 \\-1 \\0\end{array}\right] + \left[\begin{array}{rrr}0 \\0 \\0\end{array}\right] + \left[\begin{array}{rrr}2 \\1 \\0\end{array}\right] | \left[\begin{array}{rrr}2 \\-2 \\0\end{array}\right] + \left[\begin{array}{rrr}3 \\2 \\1\end{array}\right] + \left[\begin{array}{rrr}10 \\5 \\0\end{array}\right] \end{array}\right] \] \[ = \left[\begin{array}{rrr}\left[\begin{array}{rrr}4+3+6 \\-4+2+3 \\0+1+0\end{array}\right] |\left[\begin{array}{rrr}1+0+2 \\-1+0+1 \\0+0+0\end{array}\right] | \left[\begin{array}{rrr}2+3+10 \\-2+2+5 \\0+1+0\end{array}\right] \end{array}\right] \] \[ = \left[\begin{array}{rrr}13 & 3 & 15 \\1 & 0 & 5 \\1 & 0 & 1\end{array}\right]] \]

(b) Using MM (matrix multiplication)

\[ \mathbf{AxB} = \left[\begin{array}{rrr}AB_1 | AB_2 | AB_3\end{array}\right] \] \[ = \left[\begin{array}{rrr}A \left[\begin{array}{rrr}4 \\1 \\3\end{array}\right] | A\left[\begin{array}{rrr}1 \\0 \\1\end{array}\right] | A\left[\begin{array}{rrr}2 \\1 \\5\end{array}\right] \end{array}\right] \] \[ = \left[\begin{array}{rrr}\left[\begin{array}{rrr}1 & 3 & 2 \\-1 & 2 & 1 \\0 & 1 & 0\end{array}\right] \left[\begin{array}{rrr}4 \\1 \\3\end{array}\right] | \left[\begin{array}{rrr}1 & 3 & 2 \\-1 & 2 & 1 \\0 & 1 & 0\end{array}\right]\left[\begin{array}{rrr}1 \\0 \\1\end{array}\right] | \left[\begin{array}{rrr}1 & 3 & 2 \\-1 & 2 & 1 \\0 & 1 & 0\end{array}\right]\left[\begin{array}{rrr}2 \\1 \\5\end{array}\right] \end{array}\right] \] \[ = \left[\begin{array}{rrr}\left[\begin{array}{rrr}(1*4) + (3*1) + (2*3) \\(-1*4) + (2*1) + (1*3) \\(0*4) + (1*1) + (0*3)\end{array}\right] | \left[\begin{array}{rrr}(1*1) + (3*0) + (2*1) \\(-1*1) + (2*0) + (1*1) \\(0*1) + (1*0) + (0*1)\end{array}\right] | \left[\begin{array}{rrr}(1*2) + (3*1) + (2*5) \\(-1*2) + (2*1) + (1*5) \\(0*2) + (1*1) + (0*5)\end{array}\right]\end{array}\right] \] \[ = \left[\begin{array}{rrr}13 & 3 & 15 \\1 & 0 & 5 \\1 & 0 & 1\end{array}\right] \]