To find the determinant of the expression \(A^{99} - A^{98}\), where \(A = \left( {\begin{array}{} 9&6\\ 8&7 \end{array}} \right)\), we can utilize properties of determinants and matrix operations.

Determinant Calculation Steps

First, we calculate the determinant of the given matrix A. The determinant of a 2x2 matrix \( \left( {\begin{array}{} a & b \\ c & d \end{array}} \right) \) is calculated using the formula \( ad - bc \).

For matrix A:

\(\det(A) = (9 \times 7) - (6 \times 8)\)

\(\det(A) = 63 - 48\)

\(\det(A) = 15\)

Matrix Expression Simplification

Next, we simplify the expression \(A^{99} - A^{98}\). We can factor out the common term \(A^{98}\):

\(A^{99} - A^{98} = A^{98}(A - I)\)

Here, \(I\) represents the identity matrix of the same dimension as A, which is \(I = \left( {\begin{array}{} 1&0\\ 0&1 \end{array}} \right)\).

Calculating Determinant of (A - I)

Now, we compute the matrix \(A - I\):

\(A - I = \left( {\begin{array}{} 9&6\\ 8&7 \end{array}} \right) - \left( {\begin{array}{} 1&0\\ 0&1 \end{array}} \right)\)

Subtracting the corresponding elements:

\(A - I = \left( {\begin{array}{} 9-1 & 6-0\\ 8-0 & 7-1 \end{array}} \right)\)

\(A - I = \left( {\begin{array}{} 8&6\\ 8&6 \end{array}} \right)\)

The determinant of this resulting matrix \(A - I\) is:

\(\det(A - I) = (8 \times 6) - (6 \times 8)\)

\(\det(A - I) = 48 - 48\)

\(\det(A - I) = 0\)

Final Determinant Calculation

To find the determinant of the entire expression \(A^{98}(A - I)\), we use the property \(\det(XY) = \det(X)\det(Y)\). This gives us:

\(\det(A^{98}(A - I)) = \det(A^{98}) \times \det(A - I)\)

We also use the property \(\det(A^n) = (\det(A))^n\). So,

\(\det(A^{98}) = (\det(A))^{98} = (15)^{98}\)

Substituting the determinants we found:

\(\det(A^{99} - A^{98}) = (15)^{98} \times 0\)

\(\det(A^{99} - A^{98}) = 0\)

Therefore, the determinant of \(A^{99} - A^{98}\) is 0.