What is the inverse of A = ((-5 4) (0 4)) and what is the main diagonal of A = ((0 6 2) (2 6 9))?

Steps to Solve

1. Finding the Inverse of Matrix A

To find the inverse of the matrix

$$ A = \begin{pmatrix} -5 & 4 \ 0 & 4 \end{pmatrix}, $$

we use the formula for the inverse of a 2x2 matrix:

$$ A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}, $$

where ( a, b, c, d ) are elements from the matrix. Here, ( a = -5, b = 4, c = 0, d = 4 ).

First, we calculate the determinant ( ad - bc ):

$$ ad - bc = (-5)(4) - (4)(0) = -20. $$

Now plug in the values:

$$ A^{-1} = \frac{1}{-20} \begin{pmatrix} 4 & -4 \ 0 & -5 \end{pmatrix} = \begin{pmatrix} -\frac{1}{5} & \frac{1}{5} \ 0 & -\frac{1}{4} \end{pmatrix}. $$

2. Identifying the Main Diagonal Elements of Matrix A

The main diagonal elements of the matrix

$$ A = \begin{pmatrix} 0 & 6 & 9 \ 2 & 6 & 7 \end{pmatrix} $$

are the elements where the row and column indices are the same.

The main diagonal elements are:

• First element: ( A_{11} = 0 )
• Second element: ( A_{22} = 6 )

Since this matrix is not square and does not have a typical main diagonal, we will consider the elements in possibilities given in the choices.

Thus we check the options:

• Option (a): ( (4, 6, 8) )
• Option (b): ( (2, 6, 2) )
• Option (c): ( (-4, 6, 8) )

Since no option reflects the diagonal elements for a rectangular matrix, we take the given valid choices.

3. Identifying the Correct Answer for the Main Diagonal

The only correct representation for a valid rectangular case can give us ( 2 ) as a valid element repeated.