Let M,N and K be matrices of type 3×2,2×3 and 3×3 respectively, such that det...

Step 1
Determinant properties help to find the determinant of the given matrix.
Determinant property for the multiples helps to do the required, which is defined as ${\displaystyle |k\cdot A|\mathrm{\%}\left\{n\right\}=k^n\cdot {\left|A\right|}^n}$.
Here k is the real number whereas A is the square matrix.
The determinant of the matrix is equal to the determinant of its transpose.
Step 2
Apply the determinant properties for the multiples to find the required answer.
Apply the determinant properties for the product of matrices, which is defined as ${\displaystyle |A\cdot B|=\left|A\right|\cdot \left|B\right|}$.
Put the values of the determinants in equation (1) and solve.
${\displaystyle {det\left[{(-3MN)}^T{K}^{-1}\right]}^{-1}={(-3)}^{-1}{\left(det\left(MN\right)\right)}^T-1{det\left(K\right)}^{-1}-1=-13\left(det\left(MN\right)T\right)-1det\left(K\right)\dots \dots \dots \dots \dots \dots \dots ..\left(1\right)=-13\left(2\right)-1\left(6\right)=-13126=-1}$