Calculating the Determinant
Determinants of Special Matrices
Determinants of Manipulated Matrices
Invertibility
Cramer's Rule How to Examples Exercise
How to:
In this section, we're going to look at what happens when we manipulate matrices, such as in the steps of Gauss-Jordan Elimination.
Let's assume we have an NxN matrix, A = [v1 v2 v3 .. vN], where the v#'s are 1xN column vectors. What happens if we multiply one of the columns, say v2, by a scalar k:
det([v1 k*v2 v3 ... vN]) = k*det([v1 v2 ... vN])
That is to say, changing a column by a scalar multiple, also changes the determinant by a scalar multiple.
Now, let's assume A is composed of Nx1 row vectors, that is A = [v1; v2; ...; vN].
In performing Gauss-Jordan, we manipulate a matrix in three different ways:
1) Divide a row by a scalar, k
2) Swap rows
3) Add a scalar multiple of one row to another
In all of these cases, the determinant of the new matrix is easily calculated knowing the determinant of the old matrix:
| Operation Performed | New Value of Determinant |
|---|---|
vI = vI/k | |
vI = vI + k*vJ |
So, in Case (1) above, the determinant is also divided by k; in Case (2) we get the negative of our current determinant; and in Case (3) the determinant is unchanged.
Examples:
Here will be a good review of Gauss-Jordan Elimination. Starting with the matrix A = [3 6 6; 2 4 17; 2 6 14], whose determinant is det(A) = -78, let's put it into RREF, step-by-step, and keep track of the determinant along the way.
| Step performed | New determinant |
|---|---|
| A(1, :) = A(1, :)/3 | det(A)/3 = -26 |
| A(2, :) = A(2, :) - 2*A(1, :) | det(A) = -26 |
| A(3, :) = A(3, :) - 2*A(1, :) | det(A) = -26 |
| A = A([1, 3, 2], :) | -det(A) = 26 |
| A(2, :) = A(2, :)/2 | det(A)/2 = 13 |
| A(1, :) = A(1, :) - 2*A(2, :) | det(A) = 13 |
| A(3, :) = A(3, :)/6 | det(A)/13 = 1 |
| A(1, :) = A(1, :) + 8*A(3, :) | det(A) = 1 |
| A(2, :) = A(2, :) - 5*A(3, :) | det(A) = 1 |
Our resulting matrix is simply the identity matrix, which we know always has a determinant of 1. But, keeping track of the determinant, step-by-step, as it changed, also gave us a result of 1.
Exercise: