Answer:
1. The matrix A isn't the inverse of matrix B.
2. |B|=12, |A|=12
Step-by-step explanation:
1. We want to know if matrix A is the inverse of matrix B, this means that if you do the product between B and A you have to obtain the identity matrix.
We have:
[tex]A=\left[\begin{array}{cc}4&-2\\-1&3\end{array}\right][/tex]
and
[tex]B=\left[\begin{array}{cc}3&2\\1&4\end{array}\right][/tex]
A and B are 2×2 matrices (2 rows and 2 columns), if you multiply them you have to obtain a 2×2 matrix.
Then if A is the inverse of B:
[tex]B.A=I[/tex]
Where,
[tex]I=\left[\begin{array}{cc}1&0\\0&1\end{array}\right][/tex]
Observation:
If you have two matrices:
[tex]A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\\and\\B=\left[\begin{array}{cc}e&f\\g&h\end{array}\right]\\\\\\A.B=\left[\begin{array}{cc}(a.e+b.g)&(a.f+b.h)\\(c.e+d.g)&(c.f+d.h)\end{array}\right][/tex]
Now:
[tex]B.A=\left[\begin{array}{cc}3&2\\1&4\end{array}\right].\left[\begin{array}{cc}4&-2\\-1&3\end{array}\right]\\\\\\B.A=\left[\begin{array}{cc}4.3+(-2).1&4.2+(-2).4\\(-1).3+3.1&(-1).2+3.4\end{array}\right]\\\\\\B.A=\left[\begin{array}{cc}12-2&8-8\\-3+3&-2+12\end{array}\right]\\\\\\B.A=\left[\begin{array}{cc}10&0\\0&10\end{array}\right][/tex]
[tex]B.A=\left[\begin{array}{cc}10&0\\0&10\end{array}\right]\neq \left[\begin{array}{cc}1&0\\0&1\end{array}\right]=I\\\\\\B.A\neq I[/tex]
Then, the matrix A isn't the inverse of matrix B.
2. If you have a matrix A:
[tex]A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right][/tex]
The determinant of the matrix is:
[tex]|A|=ad-bc[/tex]
Then the determinant of B is:
[tex]B=\left[\begin{array}{cc}3&2\\1&4\end{array}\right][/tex]
[tex]a=3, b=2, c=1, d=4[/tex]
[tex]|B|=3.4-2.1\\|B|=12-2=10[/tex]
The determinant of A is:
[tex]A=\left[\begin{array}{cc}4&-2\\-1&3\end{array}\right][/tex]
[tex]a=4, b=-2, c=-1, d=3[/tex]
[tex]|A|=4.3-(-2).(-1)\\|B|=12-2=10[/tex]
