Let A=3a-125cb82 is symmetric and B=d3ab-ae-2b-c-26-f is skew-symmetric, find AB.

A
$A B = [\begin{array}{ccc} 4 & - 3 & 6 \\ - 31 & 54 & - 26 \\ - 28 & 9 & - 50 \end{array}]$
B
$A B = [\begin{array}{ccc} - 4 & 3 & - 6 \\ - 31 & 54 & - 26 \\ - 28 & 9 & - 50 \end{array}]$
C
$A B = [\begin{array}{ccc} 4 & 3 & - 6 \\ 31 & 54 & 26 \\ - 28 & 9 & - 50 \end{array}]$
D
$A B = [\begin{array}{ccc} 4 & 3 & 6 \\ 31 & - 54 & 26 \\ - 28 & 9 & - 50 \end{array}]$

Solution:

$∵ A$ is symmetric

$∴ c = 8, b = - 1 a n d a = 2 a n d B$ is skew-symmetric ..(i)

$∴ d = e = f = 0 a n d 2 b + c = 6, a = 2, b - a = - 3$ ..(ii)

From Eqs. (i) and (ii), we get

$a = 2, b = - 1, c = 8, d = 0, e = 0, f = 0$

$∴ A = [\begin{matrix} 3 & 2 & - 1 \\ 2 & 5 & 8 \\ - 1 & 8 & 2 \end{matrix}]$ and $B = [\begin{matrix} 0 & 3 & 2 \\ - 3 & 0 & - 6 \\ - 2 & 6 & 0 \end{matrix}]$

$\Rightarrow A B = [\begin{matrix} - 4 & 3 & - 6 \\ - 31 & 54 & - 26 \\ - 28 & 9 & - 50 \end{matrix}]$

Let $A = [\begin{matrix} 3 & a & - 1 \\ 2 & 5 & c \\ b & 8 & 2 \end{matrix}]$ is symmetric and $B = [\begin{matrix} d & 3 & a \\ b - a & e & - 2 b - c \\ - 2 & 6 & - f \end{matrix}]$ is skew-symmetric, find $A B$ .

Solution:

Related content