Question

A = ???? 4 −2

Answer 1

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:

$A=\left[\begin{array}{cc}4&-2\\-1&3\end{array}\right]$

and

$B=\left[\begin{array}{cc}3&2\\1&4\end{array}\right]$

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:

$B.A=I$

Where,

$I=\left[\begin{array}{cc}1&0\\0&1\end{array}\right]$

Observation:

If you have two matrices:

$A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\\and\\B=\left[\begin{array}{cc}e&f\\g&h\end{array}\right]\\\\\\A.B=\left[\begin{array}{cc}(a.e+b.g)&(a.f+b.h)\\(c.e+d.g)&(c.f+d.h)\end{array}\right]$

Now:

$B.A=\left[\begin{array}{cc}3&2\\1&4\end{array}\right].\left[\begin{array}{cc}4&-2\\-1&3\end{array}\right]\\\\\\B.A=\left[\begin{array}{cc}4.3+(-2).1&4.2+(-2).4\\(-1).3+3.1&(-1).2+3.4\end{array}\right]\\\\\\B.A=\left[\begin{array}{cc}12-2&8-8\\-3+3&-2+12\end{array}\right]\\\\\\B.A=\left[\begin{array}{cc}10&0\\0&10\end{array}\right]$

$B.A=\left[\begin{array}{cc}10&0\\0&10\end{array}\right]\neq \left[\begin{array}{cc}1&0\\0&1\end{array}\right]=I\\\\\\B.A\neq I$

Then, the matrix A isn't the inverse of matrix B.

2. If you have a matrix A:

$A=\left[\begin{array}{cc}a&b\\c&d\end{array}\right]$

The determinant of the matrix is:

$|A|=ad-bc$

Then the determinant of B is:

$B=\left[\begin{array}{cc}3&2\\1&4\end{array}\right]$

$a=3, b=2, c=1, d=4$

$|B|=3.4-2.1\\|B|=12-2=10$

The determinant of A is:

$A=\left[\begin{array}{cc}4&-2\\-1&3\end{array}\right]$

$a=4, b=-2, c=-1, d=3$

$|A|=4.3-(-2).(-1)\\|B|=12-2=10$