Question

Evaluate \( \operatorname{det}(A) \) by a cofactor expansion along a row or column of your choice. \[ A=\left[\begin{array}{ccc} k+1 & k-1 & 5 \\ 2 & k-10 & 5 \\ 6 & k+1 & k \end{array}\right] \] NOTE

Answer 1

The determinant of matrix A is calculated by performing a cofactor expansion along the first column. By calculating the minors of each element in the first column and their corresponding cofactors, we obtain the formula for the determinant which involves performing certain multiplications and combining like terms.

Step-by-step explanation:

To evaluate the determinant of matrix A using a cofactor expansion, we can choose any row or column. Here, we'll choose the first column, as it contains the variable k only once, which may simplify our calculations.

We start the cofactor expansion along the first column of A:

• The minor of the element a11 (k+1) is the determinant of the 2x2 matrix obtained by deleting the first row and first column from A. This minor is (k-10)*k - (5)*(k+1).
• The minor of the element a21 (2) is the determinant of the 2x2 matrix obtained by deleting the second row and first column from A. This minor is (k-1)*k - (5)*(6).
• The minor of the element a31 (6) is the determinant of the 2x2 matrix obtained by deleting the third row and first column from A. This minor is (k-1)*(k-10) - (5)*(2).

Now, we combine these minors with their corresponding cofactors to obtain the determinant of A:

det(A) = (k+1)((k-10)k - (5)(k+1)) - 2((k-1)k - 30) + 6((k-1)(k-10) - 10)
To find the exact value of the determinant of A, we need to perform the multiplications and combine like terms accordingly.