Final answer:
The matrix is expanded along row one, column three by multiplying the element in that position (which is 1) with its corresponding minor determinant (-11), resulting in an expansion value of -11.
Step-by-step explanation:
Expanding the Matrix Using Row One, Column Three
To expand the matrix using row one, column three, you use the matrix element in that position, which is 1, and its corresponding minor. Here, expanding along this element means calculating the determinant of the 2x2 matrix that remains after removing the row and column that the element is in. In this case, the remaining 2x2 matrix is:
| 4 3 |
| 1 -2 |
To find the minor, we calculate the determinant of this matrix:
(4)(-2) - (3)(1) = -8 - 3 = -11
Since our element is in the third column of the first row, we multiply it by (-1)^(1+3) to get the cofactor:
1 * (-1)^4 * (-11) = 1 * 1 * (-11) = -11
The expanded matrix along row one, column three is the product of the element and its minor:
-11