Alyce Wilkinson

2021-08-11

Let M,N and K be matrices of type $3\times 2,2\times 3$ and $3\times 3$ respectively, such that det (MN) $=2$ and $det\left(K\right)=6$ . Then which of the following is the value of the det $\left[{(-3MN)}^{T}{K}^{-1}\right]}^{-1$ ?

a)$-{3}^{2}.2$ b)$\frac{-1}{{3}^{3}.2}$ c) $\frac{2}{{3}^{2}}$ d)$\frac{-{3}^{2}}{4}$ e) $\frac{-1}{{3}^{2}}$

a)

Aamina Herring

Skilled2021-08-12Added 85 answers

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$|k\cdot A|\mathrm{\%}\left\{n\right\}={k}^{n}\cdot {\left|A\right|}^{n}$ .

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$|A\cdot B|=\left|A\right|\cdot \left|B\right|$ .

Put the values of the determinants in equation (1) and solve.

${det\left[{(-3MN)}^{T}{K}^{-1}\right]}^{-1}={(-3)}^{-1}{\left(det\left(MN\right)\right)}^{T}-1{det\left(K\right)}^{-1}-1=-13\left(det\left(MN\right)T\right)-1det\left(K\right)\dots \dots \dots \dots \dots \dots \dots ..\left(1\right)=-13\left(2\right)-1\left(6\right)=-13126=-1$

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

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

Put the values of the determinants in equation (1) and solve.