Answer to Question #201446 in Linear Algebra for Rebecca

Question

a. LCompute the product AB for

A = 0 4 0 2 3 1 3 0 1 and B = 1 0 3 1 1 5 2 3 -1

b.) Use your answer in a.) to evaluate det(AB) and compare it to det(A) det(B)

c.) Determine whether or not if det(A+B) is related to det(A) + det(B).

Answer 1

(a)

$AB=\begin{pmatrix} 0 & 4 & 0 \\ 2 & 3 & 1 \\ 3 & 0 &1 \end{pmatrix}\cdot\begin{pmatrix} 1 & 0 & 3 \\ 1 & 1 & 5 \\ 2 & 3 & -1 \end{pmatrix}$

$=\begin{pmatrix} 4 & 4 & 20 \\ 7 & 6 & 20 \\ 5 & 3 &8 \end{pmatrix}$

(b)

$\det(AB)=\begin{vmatrix} 4 & 4 & 20 \\ 7 & 6 & 20 \\ 5 & 3 &8 \end{vmatrix}$

$=4\begin{vmatrix} 6 & 20 \\ 3 & 8 \end{vmatrix}-4\begin{vmatrix} 7 & 20 \\ 5 & 8 \end{vmatrix}+20\begin{vmatrix} 7 & 6 \\ 5 & 3 \end{vmatrix}$

$=4(48-60)-4(56-100)+20(21-30)=-52$

$\det(A)=\begin{vmatrix} 0 & 4 & 0 \\ 2 & 3 & 1 \\ 3 & 0 &1 \end{vmatrix}$

$=0\begin{vmatrix} 3 & 1 \\ 0 & 1 \end{vmatrix}-4\begin{vmatrix} 2 & 1 \\ 3 & 1 \end{vmatrix}+0\begin{vmatrix} 2 & 3 \\ 3 & 0 \end{vmatrix}$

$=0-4(2-3)+0=4$

$\det(B)=\begin{vmatrix} 1 & 0 & 3 \\ 1 & 1 & 5 \\ 2 & 3 & -1 \end{vmatrix}$

$=1\begin{vmatrix} 1 & 5 \\ 3 & -1 \end{vmatrix}-0\begin{vmatrix} 1 & 5 \\ 2 & -1 \end{vmatrix}+3\begin{vmatrix} 1 & 1 \\ 2 & 3 \end{vmatrix}$

$=1(-1-15)-0+3(3-2)=-13$

$\det(AB)=-52=\det(A)\det(B)$

(c)

$A+B=\begin{pmatrix} 0 & 4 & 0 \\ 2 & 3 & 1 \\ 3 & 0 &1 \end{pmatrix}+\begin{pmatrix} 1 & 0 & 3 \\ 1 & 1 & 5 \\ 2 & 3 & -1 \end{pmatrix}$

$=\begin{pmatrix} 1 & 4 & 3 \\ 3 & 4 & 6 \\ 5 & 3 & 0 \end{pmatrix}$

$\det(A+B)=\begin{vmatrix} 1 & 4 & 3 \\ 3 & 4 & 6 \\ 5 & 3 & 0 \end{vmatrix}$

$=1\begin{vmatrix} 4 & 6 \\ 3 & 0 \end{vmatrix}-4\begin{vmatrix} 3 & 6 \\ 5 & 0 \end{vmatrix}+3\begin{vmatrix} 3 & 4 \\ 5 & 3 \end{vmatrix}$

$=-18-4(-30)+3(9-20)=69$

$\det(A+B)\not=\det(A)+\det(B)$