3 2 -1
Matrix A = 0 4 -6
2 -1 3
i)Expand and evaluate detA using minors of the first row elementary.
ii) Write down the expansion of detA using cofactors of the second column elements.
| 3 2 -1 |
| 0 4 -6 |
| 2 -1 3 |
a) 3(12 - 6) - 2(0+12) + (-1)(0 - 8) = ....
b) remember that the signs go +, -, +, -, etc
so how does the 2nd row start?
The + and - alternate in both directions.
Time to review the section, yeah?
To evaluate the determinant of matrix A, we have two methods: expanding using minors of the first row elementary and expanding using cofactors of the second column elements. Let's solve each part step by step:
i) Expanding and evaluating detA using minors of the first row elementary:
The determinant of a 3x3 matrix A can be expanded as follows:
detA = a11C11 - a12C12 + a13C13
where a11, a12, a13 are the elements of the first row of matrix A,
C11, C12, C13 are the cofactors of a11, a12, a13, respectively.
To calculate the cofactors, we will follow these steps:
1. Find the minor M11 for a11 (the determinant of the 2x2 matrix obtained by removing the first row and first column).
2. Find the minor M12 for a12 (the determinant of the 2x2 matrix obtained by removing the first row and second column).
3. Find the minor M13 for a13 (the determinant of the 2x2 matrix obtained by removing the first row and third column).
First, let's calculate the minors M11, M12, and M13:
a11 = 3
M11 = |-1 3| = (-1)(3) = -3
| 4 -6|
a12 = 2
M12 = | 0 -6| = (0)(-6) = 0
| 2 3|
a13 = -1
M13 = | 0 4| = (0)(4) = 0
| 2 -1|
Now let's calculate the cofactors C11, C12, and C13:
C11 = (-1)^2 * M11 = 1 * (-3) = -3
C12 = (-1)^3 * M12 = -1 * 0 = 0
C13 = (-1)^4 * M13 = 1 * 0 = 0
Finally, we can calculate the determinant detA:
detA = a11C11 - a12C12 + a13C13
= 3(-3) - 2(0) - (-1)(0)
= -9
Therefore, detA = -9.
ii) Writing down the expansion of detA using cofactors of the second column elements:
The expansion using cofactors of the second column elements can be written as:
detA = a21C21 - a22C22 + a23C23
where a21, a22, a23 are the elements of the second row of matrix A,
C21, C22, C23 are the cofactors of a21, a22, a23, respectively.
To calculate the cofactors, you will follow similar steps as in part i:
1. Find the minor M21 for a21 (determinant of the 2x2 matrix obtained by removing the second row and first column).
2. Find the minor M22 for a22 (determinant of the 2x2 matrix obtained by removing the second row and second column).
3. Find the minor M23 for a23 (determinant of the 2x2 matrix obtained by removing the second row and third column).
a21 = 0
M21 = |-1 3| = (-1)(3) = -3
| 2 -1|
a22 = 4
M22 = | 3 3| = (3)(3) = 9
| 2 -1|
a23 = -6
M23 = | 3 3| = (3)(3) = 9
| 2 -6|
Now let's calculate the cofactors C21, C22, and C23:
C21 = (-1)^3 * M21 = -1 * (-3) = 3
C22 = (-1)^4 * M22 = 1 * 9 = 9
C23 = (-1)^5 * M23 = -1 * 9 = -9
Finally, we substitute these values into the expansion formula:
detA = a21C21 - a22C22 + a23C23
= 0(3) - 4(9) - (-6)(-9)
= 0 - 36 + 54
= 18
Therefore, detA = 18.
So the answers to the given questions are:
i) detA = -9
ii) detA = 18