The answer is, |A| = 2, |B| = -12, and |AB| = -24.
We are given matrices A and B such that
A = $$\left[\begin{array}{rrr} -1 & 3 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]$$ and
B = $$\left[\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 6 \end{array}\right]$$,
We have to find |A|, |B|, AB, and |AB| and verify that |A||B| = |AB|.
Calculating |A|:
We know that |A| is the determinant of matrix A.
Calculating the determinant of A using the cofactor method, we get:
$$\begin{aligned}\left|\begin{array}{rrr} -1 & 3 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right|&=(-1)\left|\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right| -3\left|\begin{array}{cc} 1 & 1 \\ 0 & 0 \end{array}\right| +1\left|\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right|\\&=(-1)(-1)-3(0)+1(1)\\&=2\end{aligned}$$
Thus, |A| = 2.
Calculating |B|:
We know that |B| is the determinant of matrix B.
Calculating the determinant of B using the diagonal method, we get:
$$|B|=(-1)(2)(6)=-12$$
Thus, |B| = -12.
Calculating AB:
Multiplying matrices A and B using the matrix multiplication rule, we get:
$$\begin{aligned}AB&=\left[\begin{array}{rrr} -1 & 3 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right]\left[\begin{array}{rrr} -1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 6 \end{array}\right]\\&=\left[\begin{array}{ccc} (-1)(-1)+(3)(0)+(1)(0) & (-1)(0)+(3)(2)+(1)(0) & (-1)(0)+(3)(0)+(1)(6) \\ (1)(-1)+(0)(0)+(1)(0) & (1)(0)+(0)(2)+(1)(0) & (1)(0)+(0)(0)+(1)(6) \\ (0)(-1)+(1)(0)+(0)(0) & (0)(0)+(1)(2)+(0)(0) & (0)(0)+(1)(0)+(0)(6) \end{array}\right]\\&=\left[\begin{array}{rrr} 1 & 6 & 6 \\ -1 & 0 & 6 \\ 0 & 2 & 0 \end{array}\right]\end{aligned}$$
Thus, AB = $$\left[\begin{array}{rrr} 1 & 6 & 6 \\ -1 & 0 & 6 \\ 0 & 2 & 0 \end{array}\right]$$.
Calculating |AB|:
We know that |AB| is the determinant of matrix AB.
Calculating the determinant of AB using the cofactor method, we get:
$$\begin{aligned}\left|\begin{array}{rrr} 1 & 6 & 6 \\ -1 & 0 & 6 \\ 0 & 2 & 0 \end{array}\right|&=(1)\left|\begin{array}{cc} 0 & 6 \\ 2 & 0 \end{array}\right| -6\left|\begin{array}{cc} -1 & 6 \\ 0 & 0 \end{array}\right| +6\left|\begin{array}{cc} -1 & 0 \\ 0 & 2 \end{array}\right|\\&=(-1)(12)-(6)(0)+6(-2)\\&=-24\end{aligned}$$
Thus, |AB| = -24.
Verifying |A||B| = |AB|
We know that |A||B| = |AB| if both sides are equal.
Thus, |A||B| = |AB| = 2(-12) = -24.
Hence, |A||B| = |AB| is verified.